UC-NRU- 


IN  MEMORIAM 
FLORIAN  CAJOR1 


THE 

JUVENILE  ARITHMETICK 

AND  SCHOLARS'   GUIDE, 

[LLUSTRATED    WITH    FAMILIAR    QUESTIONS. 
AND  CONTAINING    NUMEROUS    EXAMPLES    IN 

FEDERAL,  MONEY* 

TO   WHICH  IB   ADDED,  A  8FHORT   8TUTEM  OP 

BOOK  KEEPING: 


BY  MARTIN   RTTTER, 

President  of  Augusta  College. 


REVISED  AND   ENLARGED 


By  JVathan  Guilford. 


CINCINNATI: 

FTBLI8HED  AND  SOLD   BY   N.  4&  G.  GTTILFORD. 

efcereotyped  at  the  Cincinnati  gtereotype  foundry: 

1831. 


DISTRICT  OF  OHIO,  TO  WIT : 

BE  IT  REMEMBERED,  That  on  the  twenty-second  day  of 
Vpril,  in  the  year  of  our  Lord  one  thousand  eight  hundred  and  tweii- 
y-seven,  and  in  the  fifty-first  year  of  the  American  Independence, 
VIARTIN  RUTER,  of  said  District,  hath  deposited  in  said  office  the 
itle  of  a  book,  the  right  whereof  he  claims  as  author  and  proprietor 
n  the  words  following,  to  wit: 

THE  JUVENILE  ARITHMETICK  AND  SCHOLAR'S 
GUIDE:  wherein  theory  and  practice  are  combined  and  adapted  to 
the  capacities  of  young  beginners;  containing  a  due  proportion  of 
examples  in  Federal  Monty,  and  the  whole  being-  illustrated  by  nu- 
merous questions  similar  to  those  of  PESTALOZZI,  by  J\lAR- 
TLVRUTER.A.M;" 

In  conformity  to  the  Act  of  the  Congress  of  the  United  State?,  en 
Jtled  "  An  Act  for  the  encouragement  of  Learning,  bv  securing  the 
copies  of  Maps,  Charts,  and  Books,  to  the  Authors  and  Proprietors  of 
such  copies  during  the  times  therein  mentioned  ;  and,  pl-o,  ot 
of  the  Act  entitled  "  An  Act  supplementary  to  an  Act  entitled  an  Act 
for  the  encouragement  of  Learning,  by  securing  the  copies  of  Maps, 
Charts,  and  Books,  to  the  Authors  and  Proprietors  of  such  Copies 
during  the  times  therein  mentioned,  and  extending  the  benefits  there- 
of to  the  Arts  of  designing,  engraving,  and  etching  historical  and 
other  Prints. 

\VM.  KEY  BOND, 
Clerk  of  the  District  ofOkio. 


RECOMMENDATIONS. 

The  following  have  been  selected  from  the  recommenda- 
tions bestowed  upon  this  work. 

Messrs.  Guilfords, — I  have  examined  hastily  the  "Juvenile  Arith- 
metick,"  which  you  sent  me,  and  am  of  opinion  that  it  possesses  scn»e 
advantages  over  those  generally  in  use:  I  particularly  refer  to  the 
part  intended  to  cultivate  in  the  learner,  the  habit  of  going  through 
the  solutions  mentally.  Very  respectfully  yours, 

JOHN  E.  ANNAN, 

Professor  of  Mathematicks  and  Natural  Philosophy  in  the  Miami 
University. 

Oxford,  June  5, 1827. 

From  a  hasty  review  of  Dr.  Ruter's  Artihmetick,  I  am  inclined  to 
think  well  of  it.  The  attempt  to  introduce  a  rational  method  of  in- 
struction in  any  department  of  education,  is  laudible  and  especially 
in  common  schools.  This  I  think  the  Juvenile  Arithmetick  is  well 
calculated  to  do,  in  that  branch  of  study  to  which  it  belongs.  The 
plan  of  Pestalozzi  is  excellent,  and  Dr.  Ruter  has  perhaps  imitated  it 
more  successfully  (by  comprizing  more  in  less  space)  than  Mr.  Col- 
burn,  between  whose  Arithmetick  and  this  there  is  however  a  consid- 
erable resemblance.  Your's  &c.  WM.  H.  M'GUFFY, 
une28,m7. 

Professor  of  Languages,  &c.,  in  the  Miami  University. 


We  have  examined  your  '  Juvenile  Arithmetick/  and  feel  a  pleas- 
ure in  recommending  it  to  the  schools  of  our  country.  We  think 
the  general  arrangement  good,  and  have  no  hesitation  in  saying,  that 
the  questions  prefixed  and  appended  to  the  rules,  give  it  superior  ad- 
vantages. Respectfully  yours,  JOSEPH  S.'TOMLIN  SON, 

JOHN  P.  DURBIN, 

March  12, 1828.  Professors  in  Augusta  College. 

The  Juvenile  Arithmetick,  from  the  cursory  examination  which  I 
have  given  it,  appears  to  be  a  manuel  of  value  for  the  introduction  of 
youth  into  the  science  of  numbers.  In  furnishing  a  second  edition, 
I  wish  you  success.  ELIJAH  SLACK. 

I  concur  most  cheerfully  in  the  above  opinion. 

Cincinnati,  April  2, 1828.  S.  JOHNSON. 


I  have  used  thy  compilation  of  Arithmetick  during  the  last  year; 
and  do  not  hesitate  in  recommending  it  to  the  publick.  The  ques- 
tions preceeding  the  rules,  the  particular  attention  to  fractions,  and 
the  sketch  of  mensuration  give  it  a  decided  preference  to  any  other 
here  in  use.  JOHN  L.  TALBERT. 

Cincinnati,  Fourth  mo.  5, 1828. 

Having  examined  the  above  Arithmetick,  I  cheerfully  concur  in 
the  foregoing  opinion  of  its  merits.  ARNOLD  TRUESDELL. 


i  RECOMMENDATIONS. 

I  have  carefully  inspected  the  "Juvenile  Arithmetick  and  Schol 
lar's  Guide,"  by  Dr.  Ruter,  and  am  of  the  opinion,  it  is  well  calcula- 
ted and  arranged,  to  conduct  the  pupil  by  an  easy  gradation  to  a  per- 
spicuous conception  of  the  science  of  numbers.  I  therefore  recom- 
mend it  to  the  publick  use,  particularly  in  common  schools. 

SAMUEL  BURR, 

September  2, 1827.  Professor  of  Mathematick*. 

A  cursory  examination  of  Dr.  Ruter'a  Arithmetick,  has  convinced 
me,  that  the  simple  and  familiar  manner  in  which  the  learned  author 
unfolds  the  principles  of  this  science,  and  adapts  them  to  the  under- 
standing of  the  young  learner,  can  not  fail  to  give  his  work  a  decided 
preference,  for  practical  purposes,  over  those  arithmeticks  in  common 
uge.  In  my  opinion,  teachers  who  adopt  it,  as  well  as  pupils  who  stu- 
dy it.  will  realize  satisfactory  and  highly  beneficial  results. 

S.  KIRKHAM, 

Author  of  Grammar  io  Familiar  Lectures. 
Pittiburgh,  Afril  2, 1828. 

I  have  examined  the  system  of  Arithmetick  compiled  by  Dr.  Ru- 
ter, and  am  of  opinion  that  it  is  well  calculated  tor  conveying  to 
youth,  a  general  knowledge  of  that  science  in  a  shorter  time,  man 
any  I  have  seen.  G.  GARDNER, 

March  29, 1828.    Teacher  of  Mathematicks,  Mill-Creek  Township. 


Having  examined  the  Juvenile  Jlrithmetick,  I  have  no  hesitation 
n  pronouncing  it  an  excellent  elementary  School  Book,  The  rules 
are  judiciously  arranged,  and  peculiarly  well  adapted  to  juvenile  com- 
prehension: The  work  contains  multum  in  parvo,  and  I  think  its 
publication  will  be  conducive  to  publick  utility. 

Hoping  its  merits  will  be  duly  appreciated,  1  take  great  pleasure  in 

'Commending  it  to  the  publick  patronage.     Yours  respectfully, 

RICHARD  MORECRAFT. 

Cincinnati,  January  2, 1828.  Teacher. 


From  my  acquaintance  with  Ruter'*  Arithmetick,  I  am  convinced 
that  it  is  well  calculated  to  encourage  the  student,  improve  his  mind, 
and  prepare  him  for  business.  JOHN  LOCKE, 

May  16, 1828.  Principal  of  Cincinnati  Female  Academy. 


Gentlemen — I  have  examined  with  some  attention  the  Juvenile 
Arithmetick,  &c.  by  the  Rev.  Dr.  Ruter,  and  am  decidely  of  opinion, 
that  it  is  admirably  calculated  for  conveying  to  youth  with  great  fa- 
cility a  general  knowledge  of  that  important  science.  The  ingenious 
manner  in  which  the  compiler  has  given  an  elucidation  of  Vulgar 
Fractions,  together  with  an  exclusion  of  all  extraneous  matter,  ren- 
ders it  in  my  estimation  a  treatise  of  peculiar  merit. 
Your  obedient  servant, 

JOHN  WINRIGHT, 

Cincinnati,  September  2, 1827.  Teachar. 


PREFACE. 


Tins  ARITHMETICK  has  been  compiled  with  a  view 
to  facilitate  the  progress  of  pupils,  and  lessen  the  labour 
of  teachers.  The  questions  preceding  and  following 
the  rules,  are  designed  to  lead  young  learners  into  habits 
of  thinking  and  calculating;  and  thus,  to  prepare  them 
for  practical  operations.  Experience  has  demonstrated, 
that,  in  the  instruction  of  children  in  any  science,  it  is 
necessary  to  excite  their  entire  attention  to  the  subject 
before  them.  The  latent  energies  of  their  minds  must 
be  roused  up,  and  called  forth  into  action.  When  this 
can  be  effectually  done,  success  is  rendered  certain. — 
To  accomplish  this  important  object,  the  best  method 
has  been  found  in  the  frequent  use  of  well  selected 
questions.  Though  it  is  a  successful  course  in  all  ju- 
venile studies,  it  is  particularly  so  in  the  science  of 
numbers ;  and  the  progress  of  pupils  must  be  slow  with 
out  it.  The  questions  in  the  following  pages  are 
thought  to  be  sufficiently  numerous  for  the  purposes  in- 
tended ;  the  rules  have  been  arranged  according  to  the 
plan  of  some  of  the  best  authors  on  this  subject,  ard 
the  work  is  offered  to  the  publick  with  the  hope  that  it 
will  be  useful  in  the  schools  of  our  country. 

M.R. 
1* 


EXPLANATION  OF  THE  CHARACTERS  USED 
IN  ARITHMETICS 


-[*         Signifies  plus,  or  addition. 

Signifies  minus,  or  subtraction. 
Denotes  multiplication. 
Means  division. 
: :  :   Signifies  proportion. 

Denotes  equality. 

Thus,  4-{-7  denotes  that  7  is  to  be  added  to  4. 
5 — 3,  Denotes  that  3  is  to  be  taken  from  5. 
8X2,  Signifies  that  8  is  to  be  multiplied  by  2. 
9-r-3,  That  9  is  to  be  divided  by  3. 
3:2:  :  6: 4,  Shows  that  3  is  to  2  as  6  is  to  4. 
7+9=  16,  Shows  that  the  sum  of  7  and  9  is  equal  to  16 
V  or  3 ,/  Denotes  the  Square  Root. 
*J  Denotes  the  Cube  Root. 
4  J  Denotes  the  Biquadrate  Root. 

This  mark,  called  a  Vinculum,  shows  that  the 

several  figures  over  which  it  is  drawn  are  to 
to  be  taken  together  as  a  simple  quantity. 


ARITHMETIC!*. 


ARITHMETICK  is  the  science  which  treats  of  the  nature 
and  properties  of  numbers  :  and  its  operations  are  con- 
ducted  chiefly  by  five  principal  rules.  These  are,  Nu- 
meration, Addition,  Subtraction,  Multiplication,  and  Di- 
vision. 

Numhers  in  Arithmetick  are  expressed  by  the  fol 
lowing  ten  digits  or  characters,  namely  :  1  one,  2  two, 

3  three,  4  four,  5  five,  6  six,  7  seven,  8  eight,  9  nine,  0 
cypher. 

An  Integer  signifies  a  whole  number,  or  certain  quan 
tity  of  units,  as  one,  three,  ten.  A  Fraction  is  a  broken 
number,  or  part  of  a  number,  as  i  one  half,  §  two-thirds, 

4  one-fourth,  |  three-fourths,  I  five-eigths,  &c. 

Numeration  teaches  the  different  value  of  figures  by 
their  different  places,  and  to  express  any  proposed  num 
bers  either  by  words  or  characters  ;  or  to  read  and 
write  any  sum  or  number. 


miMERATION  TABLE. 

r-HO?'NO}COTj<    Units. 
ojO'— iooTt<co    Tens. 

Hundreds. 
Thousands. 
CO^COTJ.IO.XN   Tens  of  thousands. 
01  co  ^  ifl  ^  x  eo   Hundreds  of  thousands. 
^*»«««    Millions, 
co i- woo  10    Tens  of  millions. 
00  ^  *-  »   Hundreds  of  millions. 
'   Thousands  of  millions. 
10  °°    Ten  thousands  of  millions. 
°*   Hundred  thousands  of  millions. 


8  NUMERATION. 

Here  any  figure  in  the  place  of  units,  reckoning  from 
right  to  left,  denotes  only  its  simple  value  ;  but  that  in 
the  second  place  denotes  ten  times  its  simple  value ;  and 
that  in  the  third  place,  one  hundred  times  its  simple 
value  ;  and  so  on,  the  value  of  any  figure  in  each  suc- 
cessive place,  being  always  ten  times  its  former  value. 
Thus  in  the  number  6543,  the  3  in  the  first  place  denotes 
only  three  ;  but  4  in  the  second  place  signifies  four  tens 
or  40 ;  5  in  the  third  place,  five  hundred  ;  and  six  in  the 
fourth  place,  six  thousand  ;  which  makes  the  whole 
number  read  thus — six  thousand  five  hundred  and  forty- 
three.  The  cypher  stands  for  nothing  when  alone,  or 
I  when  on  the  left  hand  side  of  an  integer  ;  but  being 
joined  on  the  right  hand  side  of  other  figures,  it  increas- 
es their  value  in  the  same  ten  fold  proportion  :  thus,  50 
denotes  Jive  tens  ;  and  500  is  read  jfoe  hundred. 

Though  the  preceding  numeration  table  contains  only 
twelve  places,  which  render  it  sufficiently  large  for 
[young  students,  yet  it  may  be  extended  to  more  places 
I  at  pleasure. 

EXAMPLE. 


Quatrillions.      Trillions.     Billions.    Millions.     Units. 
987,654  ;         321,234  ;   567,898  ;    765,432  ;  123,456 
Here  note,  that  Billions  is  substituted  for  millions  of 
millions  :  Trillions,  frr  millions  of  millions  of  millions  : 
Quatrillions,  for  millions  of  millions  of  millions  of  mil- 
lions.    From  millions,  to  billions,  trillions,  quatrillions, 
and  other  degrees  of  numeration,  the  same  intermediate 
denominations,  of  tens,  hundreds,  thousands,  fyc.  are 
'ised,  as  from  units  to  millions.     And  thus,  in  ascertain- 
ing the  amount  of  very  high  numbers,  we  proceed  from 
Millions  to  Billions,  Trillions,  Quatrillions,  Quintillions, 
SextillioRS,  Septillions,  Octillions,  Nonillions,  Decillions, 
Undecillions,  Duodecillions,  Tredecillions,  Quatuorde- 
cillions,  Quindecillions,  Sexdeciliions,  Septendecillions, 
Octodecil  lions.,  Novemdecillions,  Vigintillions,  &c.  all 
|! r)f  which  answer  to  millions  so  often  repeated,  as  their 
|j  indices  respectively  require,  according  to  the  above  pro- 
f  j  portion. 


THE  APPLICATION. 

Write  down  in  figures  the  following  numbers 
Ten.  .       )0 

Twenty-one.        .... 

Thirty-five.     -  -         -       35 

Four  hundred  and  sixty-seven.  467 

Two  thousand  three  hundred  and  eighty-nine.  -  23S9 
Thirty -four  thousand  live  hundred  and  seventy  04570 
Six  hundred  and  three  thousand  four  hundred.  6034  )C 
Seven  millions  eight  hundred  end  four  thou- )  -cn/lqr>q 

sand  three  hundred  and  twenty-nine. 
Fifty -eight  millions  seven  hundred  and  thir- 
ty-two thousand  one  hundred  and  five. 
Eight  hundred  and  ten  millions  nine  him- 


dred  and  two  thousand  five  hundred 


610902512 


and  twelve. 
Three  thousand  two  hundred  and  three 

millions  six  hundred  and  eight  thou-£  3203608999 
sand  nine  hundred  and  ninety -nine. 

Question  1.  What  is  Arithmetic!*.  ? 

2.  What  are  the  ten  digits  by  which  numbers 

are  expressed  ? 

3.  What  is  an  integer  ? 

4.  What  is  a  fraction  ? 

5.  What  are  the  principal  rules  by  which  the 

operations  in  Arithmetick  are  conducted? 
6.What  does  Numeration  teach  ? 


SIMPLE  ADDITION. 

Simple  Addition  teaches  to  put  together  numbers  of 
;he  same  denomination  into  one  sum  ;  as  5  dollars,  4 
dollars,  and  3  dollars,  make  12  dollars. 

Before  the  pupil  enters  upon  Addition  in  the  usual 
way,  with  figures,  it  would  be  useful  for  him  to  learn  to 
perform  easy  operations  in  his  mind.  For  this  purpose 
et  him  be  exercised  in  the  following  questions,  or  in 
others  which  are  similar. 


to 


SIMPLE  ADDITION. 


1 .  If  you  have  two  cents  in  one  hand  and  two  in  the 
other,  how  many  have  you  in  both  ? 

2.  If  you  have  three  cents  in  one  hand,  and  two  in 
he  other,  how  many  have  you  in  both  ? 

3.  If  you  have  five  cents  in  one  hand,  and  two  in  the 
jther,  how  many  have  you  in  both  ? 

4.  John  has  six  cents,.-  and  Robert  has  three  ;  how 
many  have  they  both  together  ? 

5.  Charles  gave  five  cents  for  an  orange,  and  two  for 
in  apple  ;  how  many  clid  he  give  for  both  ? 

6.  Dick  had  four  nuts,  John  had  three,  and  David 
!iad  two  ;  how  many  had  they  all  together  ? 

7.  Henry  had  five  peaches,  Joseph  had  three,  and 
Tom  had  two,  and  they  put  them  all  into  a  basket  ;  how 
many  were  there  in  the  basket  ? 

8.  Three  boys,  Peter  John  and  Oliver,  gave  some 
money  to  a  beggar.     Peter  gave  seven  cents,  John  four, 
ind  Oliver  three.     How  many  did  they  all  give  him  ? 

9.  A  man  bought  a  sheep  for  eight  dollars,  and  a  calf 
for  seven  dollars  ;  what  did  he  give  for  both  ? 

10.  A  boy  gave  to  one  of  his  companions  eight  peach- 
es •  to  another  six  ;  to  another  four  ;  and  kept  two  him- 
self ;  how  many  had  he  at  first  ? 

11.  How  many  are  two  and  three  ? — two  and  five  ? — 
three  and  seven  ? — four  and  five  ? 

12.  How  many  are  two  and  four  and  one? 

13.  How  many  are  three  and  two  and  one? 

14.  How  many  are  four  and  three  and  two? 

15.  How  many  are  five  and  four  and  three  ? 

16.  How  many  are  four  arid  five  and  two  ? 

17.  How  many  are  seven  and  three  and  one  ? 

18.  How  many  are  eight  and  four  and  two  ? 

19.  How  many  are  nine  and  five  and  one  ? 

20.  How  many  are  five  and  six  and  seven  ? 

21.  How  many  are  four  and  three  and  two  and  one  ? 

22.  How  many  are  two  and  three  and  one  and  four  ? 

I  23.  How  many  are  five  and  three  and  two  and  one  ?* 

*It  is  expected  that  many  of  these  questions  will  be  varied  by  the 
teacher,  and  rendered  harder,  or  easier,  or  others  substituted  as  the 
capacity  of  the  pupil  may  require. 


SIMPLE    ADDITION.  ll 

RULE. 

Place  the  figures  to  be  added,  one  under  another,  so 
that  units  will  stand  under  units,  tens  under  tens,  hun- 
dreds under  hundreds,  &c.  Draw  a  horizontal  line  un- 
der them,  and  beginning  at  the  bottom  of  the  first  col- 
umn, on  the  right  hand  side,  that  is,  at  units,  add  togeth- 
er the  figures  in  that  column,  proceeding  from  the  bot- 
tom to  the  top.  Consider  how  many  tens  are  contained 
in  their  sum,  and  how  many  remain  besides  the  even 
number  of  tens  ;  place  the  amount  under  the  column 
of  units,  and  carry  so  many  as  you  have  tens  to  the 
next  column.  Proceed  in  the  same  manner  through 
every  column,  setting  down  under  the  last  column  ih 
full  amount. 

PROOF. 

Begin  at  the  top  of  the  sum  and  add  the  Several  rows 
of  figures  downwards  as  they  were  added  upwards,  and 
it  the  additions  in  both  cases  be  correct,  the  sums  will 
agree. 

EXAMPLES. 
I.  II.  III.  IV. 

12  321  4000  542210 

21  123  3124  18540? 

34  410  2345  350212 

10  203  5234  201304 


1057    14703 


V.  VI.                VII. 

2405670  50678  4  5  0  7  8  f 

3540210  76543  876542 

4321023  20134  450780 

4065243  56787  876543 

2123456  65432  2  3  4  7  9  g 


SIMPLE  ADDITION. 

VIII.  IX,  X. 

57898765  45  20000000 

4321234  678  3000000 

567898  9876  400000 

76543  54321  50000 

2123  234567  6000 

212  987654  700 


72866775     9287141    2345670C 


XL  XII.  XIII. 

24681012  54321231  98765432 

42130538  19000310  12345576 

71021346  20304986  98765432 

20324213  19876540  12345678 

98765432  98755432  98765432 

12345678  12000987  12345076 


APPLICATION. 

1.  A  boy  owed  one  of  his  companions  6  cents  ;  he 
owed  another  8,  another  5,  and  another  9.  How  much 
did  he  owe  in  all  ?  Ans.  28  cents. 

2'.  A  man  received  of  one  of  his  friends  7  dollars, 
of  another  10,  of  another  19,  and  of  another  50.  How 
many  dollars  did  he  receive  ?  Ans.  86  dollars. 

3.  A  person  bought  of  one  merchant  ten  barrels  of 
flour,  and  paid  40  dollars  ;  of  another  20  barrels  of 

ider,  for  which  he  paid  60  dollars,  and  20  barrels  of 
sugar  at  450  dollars  ;  and  of  another  95  barrels  of  salt 
at  570.  How  many  barrels  did  he  buy,  and  how  much 
money  did  he  pay  for  the  whole  ? 

Ans.  145  barrels,  and  paid  1120  dollars. 

4.  A  had  250  dollars  ;  B  had  375  ;  C  had  5423  ;  D, 
84320  ;    E,  287432,  and  F,  4321567.      How  much 
would  it  all  make,  if  put  together  ?       Ans.  $4679367. 

Question.  1.  What  does  Simple  Addition  teach  ? 

2.  How  do  you  place  the  numbers  to  be  added? 
Where  do  you  begin  the  Addition? 
How  do  you  prove  a  sum  in  Addition? 


SIMPLE  SUBTRACTION. 

Simple  Subtraction  teaches  to  take  a  less  number 
from  a  greater  of  the  same  denomination,  and  thus  tc 
find  the  difference  between  them. 

Questions  to  prepare  the  learner  for  this  rule. 

1.  If  you  have  seven  cents,  and  give  away  two;  how 
many  will  you  have  left? 

2.  If  you  have  eight  cents,  arid  loee  four  of  them; 
how  many  will  you  have  left? 

3.  A  boy  having  ten  centa,  gave  away  four  of  them; 
how7  many  had  he  iefe? 

4.  A  man  owing  twelve  dollars,  paid  four  of  it;  how 
much  did  he  then  owe? 

5.  A  man  bought  a  firkin  of  butter  for  fifteen  dollars, 
and  sold  it  again  for  ten  dollars ;  how  much  did  he  lose  ? 

6.  If  a  horse  is  worth  ten  dollars,  and  a  cow  is  worth 
four;  how  much  more  is  the  horse  worth  than  the  cow? 

7.  Ahoy  had  eleven  apples  in  a  basket,  and  took  out 
five;  how  many  were  left? 

8.  Susan  had  fourteen  cherries,  and  ate  four  of  them; 
how  many  had  she  left? 

9.  Thomas  had  twenty  cents,  and  paid  away  five  of 
them  for  some  plums;  how  many  had  he  left? 

10.  George  is  twelve  years  old,  and  William  is  seven; 
how  much  older  is  George  than  William? 

11.  Take  four  from  eight;  how  many  will  remain? 

12.  Take  three  from  nine;  how  many  will  remain? 
18.  Take  five  from  ton;  how  many  will  remain? 

14.  Take  six  from  ten ;  how  many  will  remain? 

15.  Take  six  from  eleven;  how  many  will  remain? 

16.  Take  five  from  twelve;  how  many  will  remain? 

17.  Take  four  from  thirteen;  how  many  will  remain? 

18.  Take  six  from  fourteen;  how  many  will  remain? 
10.  Take  six  from  fifteen;  how  many  will  remain? 

20.  Take  eight  from  sixteen;  how  many  will  remain? 

21.  Take  nine  from  twelve;  how  many  will  remain? 

22.  Take  nine  from  ftv.rtesn;  how  many  will  remain? 

23.  Take  three  from  thirteen;  how  many  will  remain? 

2 


14  SIMPLE  SUBTRACTION. 

24.  Take  eight  from  seventeen ;  how  many  will  remain  ? 

25.  Take  nine  from  sixteen;  how  many  will  remain? 
28.  Take  nine  from  eighteen ;  how  many  will  remain? 

RULE. 

Place  the  larger  number  uppermost,  and  the  smaller 
one  under  it,  so  that  units  may  stand  under  units ;  tens 
under  tens;  hundreds  under  hundreds,  &c.  Draw  a  line 
underneath,  and  beginning  with  units,  subtract  the  low- 
er from  the  upper  figure,  and  set  down  the  remainder. — 
But  when  in  any  place  the  lower  figure  is  larger  than 
the  upper,  call  the  upper  one  ten  more  than  it  really  is  ; 
subtract  the  lower  figure  from  the  upper,  considering  it 
as  having  ten  added  to  it,  set  down  the  remainder^  and 
add  one  to  the  next  left  figure  of  the  lower  line,  and 
proceed  thus  through  the  whole. 
PROOF. 

Add  the  remainder  and  the  less  line  together,  and  if 
the  work  be  right,  their  sum  will  be  equal  to  the  greater 
line. 

EXAMPLES. 

I.     II.       III.        IV.         V. 
23   457    54367    73214    84201 
11   215    20154    54876    49983 


12   242    34213    18338    34218 


VI.  VII. 

9812030405321     700000000000 
6054123456789      98765432123 


VIII.         IX.  X.  XI. 

32016   98700   500612   65040032 
12045   25290   499521    7000302 


XII.  XIII.  XIV. 

974865     400000     100000000 
863757  7  1 


SIMPLE  ADDITION  AND  SUBTRACTION.  15 

APPLICATION. 

1.  A  ship's  crew  consisted  of  75  men,  21    of  whom 
died  at  sea.     How  many  arrived  safe  in  port? 

Ans.  54  men,. 

2.  A  boy  had  100  miles  to  travel,  and  went  33  miles 
the  first  day.     How  far  had  he  still  to  go? 

Ans.  67  miles, 

3.  A  tree  had  647  appies  on  it,  but  158  of  them  fell 
off.     How  many  were  there  then  remaining  on  the 
tree?  Ans.  489  apples, 

4.  A  boy  put  1000  nuts  into  a  basket  and  afterwards 
took  out  650.     How  many  were  left  in  the  basket? 

Ans.  350  nuts. 

5.  A  general  had  an  army  of  43250  men,  15342  of 
them  deserted.     How  many  remained? 

Ans.  27G08  men. 
Question  1.  What  does  Subtraction  teach? 

2.  How  do  you  place  the  larger  and  smaller 

numbers? 

3.  What  do  you  do  when  the  lower  number  is 

larger  than  the  upper  number? 

4.  How  is  a  sum  in  subtraction  proved? 

Exercises  for  the  slate  under  the  two  preceding  Rules. 

1.  I  saw  15  ladies  pass  up  street,  and  8  down  street. 
How  many  passed  both  ways?  Ans.  23  ladies. 

2.  A  boy  who  had   15  buttons  upon  his  jacket,  lost 
off  7  of  them.     How  many  were  left  on? 

Ans.  8  buttons. 

3.  A  man  bought  a  barrel  of  flour  for  10  dollars,  a 
barrel  of  molasses  for  29  dollars,  arid  a  barrel  of  rum 
for  36  dollars.     How  much  did  he  pay  for  all  the  arti- 
cles? Ans.  75  dollars. 

4.  A  man  bought  a  chaise  for  175  dollars,  and  to  pay 
for  it  gave  a  wagon   worth  37  dollars  and  the  rest  in 
money.     How  much  money  did  he  pay. 

Ans.  138  dollar? 

5.  James  bought  at  one  time  89  apples,  at  another  54, 
at  another  60,  and  at  another  75.     Hovy  many  did  he 

in  all?  Ans.  278  apples 


16  SIMl'LE  ADDITION  AND  SUBTKAC'TIOK'. 

6.  A  merchant  bought  a  piece  of   cloth  containing 
489  yards,  and  sold  355  yards.     How  many  yards  hue 
he  left?  Ans.*  124  yards 

7.  Suppose  my   neighbour  should  borrow   cf  me  a 
one  time  658  dollars,  at  another  50  dollars,  at  anothei 
3655  dollars,  and  at  another  5000  dollars;  how  much 
should  I  lend  him  in  all?  Ans.  9361  dollars 

8.  Charles  has  42  marbles  and  John  has  25.     Hou 
many  has  Charles  more  than  John?     Ans.  17  marbles 

9.  If  Charles  give  John  200  nuts,  and  James  give 
him  56,  and  Joseph  give  him  195;  how  many  will  John 
have?  Ans.  451  nuts 

10.  My  friend  owed  me  150  dollars,  but  has  paid  me 
90  dollars.     How  much  does  he  still  owe  me? 

Ans.  60  dollars, 

11.  If  you  buy  20  peaches  for  40  cents,  and  sell  15 
for  35  cents,  how  many  peaches  will  you  have  left,  and 
bow  much  will  they  cost  you? 

Ans.  5  peaches,  &  will  cost  5  cents 

12.  A  person  went  to  collect  money,  and  received 
of  one  man  90  dollars;  of  another  140  dollars;  of  an 
other  101   dollars,  and  of  another  29  dollars.     How 
much  did  he  collect  in  all?  Ans.  360  dollars. 

13.  A  man  deposited  in  bank  8752  dollars,  and  drew 
out  at  one  time  4234  dollars,  at  another  1700  dollars, 
it  another  962  dollars,  and  at  another  49  dollars.  How 
much  had  he  remaining  in  bank?      Ans.  1807  dollars. 

14.  Gen.   Washington   was  born    1732,  and  died  in 
1799.     How  old  was  he  when  he  died  ? 

Ans.  67  years. 

15.  A  man  owed   11,989  dollars.     He  paid  at  one 
ime  2875  dollars;  at  another  4243;  at  another  3000 

dollars.     How  much  did  he  still  owe? 

Ans.  1871  dollars. 

16.  A  man  travelled  till  he  found  himself  1300  miles 
rom  home.     On  his  return,  he  travelled  in  one  week 

235  miles;  in  the  next  275;  in  the  next  325,  and  in 
he  next  290.  How  far  had  he  still  to  go  before  he 
vould  reach  home?  Ans.  175  miles. 


SIMPLE  MULTIPLICATION. 

Simple  Multiplication  toadies  a  short  method  of  find- 
ing what  a  number  amounts  to  when  repeated  a  given 
number  of  times,  and  thus  performs  Addition  in  a  very 
expeditious  manner. 

I.  What  will  fjur  apples  cost  at  two  cents  a  piece? 
12.  What  must  you  give  Ibr  two  oranges,  at  six  cents 

a  piece? 

3.  What  are   two  barrels  of  flour  worth,  at  five  dol- 
lars a  barrel? 

4.  What  will  three  pounds  of  butter  come  to,  at  three 
cents  a  pound? 

5.  If  you  can  walk  four  miles   in  one  hour,  how  far 
can  you  walk  in  three  hours? 

0.  If  a  cent  will  buy  five  nuts,  how  many  nuts  will 
four  cents  buv? 

7.  What  are  two  barrels  of  cider   worth,  at  three 
dollars  a  barrel? 

8.  If  you  give  four  cents  for  a  yard  of  tape,  how 
mnny  cents  will  buv  three  yards? 

l\  If  I  pat  in  your  pocket  five  r.pples  at  three  differ- 
ent time?,  how  many  apples  will  you  have  in  your 
pocket?  How  many  are  three  times  five? 

10.  If  four  boys  have  each  four  apples,  how  many 
have  they  all?  How  many  are  four  times  four? 

II.  What  will   six  marbles   cost  at  three  cents   a 
piece  ?     How  many  are  six  times  three  ? 

12.  A  horse  has  four  legs.     How  many  legs  have 
five  horses?     How  many  are  four  times  five? 

13.  I  gave  six  boys  four  peaches  each.     How  many 
di  I  I  give  them  all?     How  many  are  six  times  four? 

14.  How  many  cents  will  buy  ten   marbles  if  one 
cost  three  cents?     How  many  are  three  times  ten? 

15.  If  I  can  walk  three  miles  in  one  hour,  how  far 
can  I  walk  in  six  hours? 


Before  entering  upon  this  Rule,  let  the  pupil  so  learn  the  follow- 
ing table,  as  to  answer  with  readiness  any  question  implied  in  it; 
after  which,  he  will  be  able  to  proceed  with  facility. 


IS 

SIMPLE 

MULTIPLICATION.                                   ; 

MULTIPLICATION 

TABLE. 

Twice 

3  times 

4  times 

5  times 

o  ume,e 

/  times  I 

Imake  2 

Imake  3 

1  make  4 

Imake  5 

Imake  £ 

Jniake  7 

2         4 

2         6 

2 

6 

2 

10 

2       IS 

2       14 

3         6 

3         £ 

3 

12 

3 

15 

3       It 

3       21 

4         8 

4       12 

4 

16 

4 

20 

4       24 

4       28 

5       10 

5       15 

5 

20 

5 

25 

5       3C 

5       35 

6       12 

6       15 

6 

24 

6 

30 

6       3€ 

6       42 

7       14 

7       21 

7 

28 

7 

35 

7       42 

7       49 

8       16 

t 

3       24 

8 

32 

8 

4C 

8     r4fc 

8       56 

9       16 

9       21 

9 

36 

9 

45 

9       54 

9       63 

10       20 

10       30 

10 

4C 

10 

5<T: 

'0       6C 

10       70 

11       22 

11       33 

11 

44 

11 

55 

11       66 

11       77 

12       24 

12       36 

12 

# 

12 

6C 

12       72 

12       84 

8  times 

9  times 

10  times 

11  times 

IxJ  times 

1  make 

8 

1  make  f. 

Imake  1C 

Imake  11 

lmakel£ 

2V        16 

2 

18 

2 

2C 

2 

22 

2         24 

3          24 

3 

27 

3 

3G 

3 

33 

3         36 

'  4          32 

4 

36 

4 

40 

4 

44 

4         46 

•   5          40 

5 

45 

5 

50 

5 

55 

5         6C 

f    6          48 

6 

54 

6 

60 

6 

66 

6         7£ 

1     7          56 

7 

63 

7 

70 

7 

77 

7         84 

1    8          64 

8 

72 

8 

80 

8 

88 

8         96 

9          72 

9 

81 

9 

90 

9 

99 

9     ioe 

10          80 

10 

90 

10 

100 

10 

110 

10       12C 

11          88 

11 

99 

11 

110 

11 

121 

11        13S 

|12          96 

12 

108 

12 

120 

12 

132 

12       144 

Though  the  foregoing  table  extends  no  farther  than 

12,  it  may 

be  easily  continued  farther;  and   if  pupils 

were  to  extend  it,  and  commit  it  to  memory,  as  far  as  30 

or  40,  it 

would  afford 

them  great  advantage  in   their 

progress. 

The  number  to 

be  multiplied  is  called  the  multipli- 

cand. 

The  number  which  multiplies  is  called  the  multiplier.* 

The  number  produced  by  the  operation  is  called  the 

product. 

*The  multiplier  and 

multiplicand  are  called  Factors. 

SIMPLE  MULTIPLICATION.  19 

CASE  I 

When  the  Multiplier  is  no  more  than  12. 
RULE. 

Place  the  greater  number,  or  multiplicand,  upper- 
most; set  the  multiplier  under  it,  and  beginning  with 
units,  multiply  all  the  figures  of  the  multiplicand  in 
succession,  carrying  one  to  the  next  figure  for  every  ten, 
and  setting  down  the  several  products,  as  in  Addition. 
The  whole  of  the  last  product  must  be  set  down. 

PROOF. 

Multiply  the  sum  by  double  the  amount  of  the  multi- 
plier, and  if  the  work  in  both  instances  bo  right,  the 
product  will  be  double  the  amount  of  the  former  pro- 
duct* 


i. 

234 
2 


n. 

3201 
3 


EXAMPLES. 
III. 

51000 
4 


rv. 

43201 
5 


v. 

354610 
6 


168        9603     204000     216005     2127660 


VI. 

453210 

7 


VII. 

3245013 

8 


VIII. 

98765432  1 
9 


3  172470 


IX.  X.  XI. 

678987654  321234567  898765432 
9  11  12 


*Multi plication  may  be  proved  by  Division;  for  if  the  product  be 
divided  by  the  multiplier,  the  quotient  will  be  the  same  as  the  multi 
plicand. 

1      2* 


20  SI.MfLF  MULTIPLICATION. 

CASE  II. 

When  the  Multiplier  is  more  than  12. 
RULE. 

Multiply  each  figure  in  the  multiplicand  by  every 
figure  in  the  multiplier,  and  place  the  first  figure  of 
each  product  exactly  under  its  multiplier;  then  add  the 
several  products  together,  and  their  sum  will  be  the 
answer. 

When  cyphers  occur  at  the  right  hand  of  either  of 
the  factors,  omit  them  in  multiplying,  and  annex  them  to 
the  right  hand  of  the  product.* 

When  the  multiplier  is  the  product  of  any  two  whole 
numbers,  the  multiplication  may  be  performed  by  mul- 
tiplying the  sum  by  one  of  them,  and  the  product  by  the 
other.  Thus,  if  24  were  to  be  multiplied  by  18,  (as  6 
times  3  make  18,)  let  it  be  multiplied  by  6,  the  product 
by  3,  and  the  answer  will  be  the  same  as  if  multiplied 
by  18. 

EXAMPLES. 
I.  II. 

43021678  8765432 C 

432  543 


86043356  26296296 C 

129065034  350617280 

1 720867  12  438271600 


18585364896         4759629576C 


in.  iv.  v. 

679100     26043     432000 
32  34      4300 


13582       104172    1296 
20373        78129    1728 


21731200    885462   1857600000 

*Multiplying  by  10,  add  a  cypher  to  the  right  hand  side  of  the 
sum,  and  it  is  done.     Thus,  let  it  be  required  to  multiply  12  by  10, 
the  product  will  be  120;    but  if  a  cypher  be  added,  it  will  bring  the 
;ult.     In   multiplying  by  100,  add  two  cyphers:  by  1000, 


same    res 
three,  <fcc. 


SIMPLE  MULTIPLICATION.  2l 

Multiply  18450  by  35. 

vi.             vn.  vm. 

18450         18450  18450 

7              5  35 


129150         92250        92250 
5  7       55350 


045 750        645750       645750 


9.  multiply    420  by       7  product        2940 

10.  "3240  9  29160 

11.  54134  18  974412 

12.  37990  24  911760 

13.  84522  54  4564188 

14.  90203  587  52949161 

15.  370456  7854  2909561424 

16.  7654876  8765  67094988140 

APPLICATION. 

1.  A  man  had  29  cows,  and  his  neighbour  had  five 
times  as  many.     How  many  had  his  neighbour? 

Ans.  145. 

2.  There  are  twelve  barrels  of  sugar,  each  contain- 
ing 256  pounds.      How  manv  pounds  do  they  all  con- 
tain? Ans.  3072. 

3.  How  far  will  a  man  travel  in  a  year,  allowing  the 
year  to  contain  365  days,  if  he  travel  40  miles  per  day? 

Ans.  14600  miles. 

4.  In  one  hogshead  are  63  gallons ; — how  many  gal- 
lons are  there  in  144  hogsheads?  Ans.  9072. 

Q.  1.  What  does  simple  Multiplication  teach? 

2.  What  is  the  number  to  be  multiplied,  called? 

3.  What  is  the  number  called  which  is  used  in  mul- 

tiplying another  number? 

4.  Are  the  multiplicand  and  multiplier  called  by 

any  other  names? 

5.  How  do  you  proceed  when  the  multiplier  is  no 

more 'than  12? 

6.  When  the  multiplier  is  more  than  12,  how  do 

you  proceed? 


22  SIMPLE  DIVISION. 

7.  What  do  you  do  when  cyphers  occur  at  the  right 

hand  of  either  of  the  factors? 

8.  How  do  you  proceed  when  the  multiplier  is  the 

product  of  two  other  numbers? 

9.  How  may  sums  in  Multiplication  be  proved? 


SIMPLE  DIVISION. 

Simple  Division  teaches  to  find  how  often  one  num- 
ber is  contained  in  another,  and  is  a  concise  way  of 
performing  several  subtractions. 

Questions  to  prepare  the  learner  for  this  rule. 

1.  James  had  4  apples  and  John  half  as  many;  how 
many  had  John? 

2.  If  two  oranges  cost  6  cents,  what  does  one  cost? 

3.  If  you  divide  8  apples  equally  between  two  boys, 
how  many  will  each  have? 

4.  What  is  one  half  of  eight? 

5.  If  you  divide  6  nuts  equally  among  3  boys,  how 
many  will  each  have? 

6.  What  is  one  third  of  six? 

7.  If  12  cherries  cost  9  cents,  what  will  4  cost? 

8.  A  third  of  9  is  how  many? 

9.  If  you  divide  16  nuts  equally  among  4  boys,  how 
many  will  each  have? 

10.  A  fourth  of  16  is  how  many  ? 

11.  How  many  times  two  are  there  in  six? 

12.  How  many  times  three  in  six? 

13.  How  many  times  four  in  eight? 

14.  How  many  times  two  in  twelve? 

15.  In  nine,  how  many  times  three? 

16.  In  eight,  how  many  times  two? 

17.  In  ten,  how  many  times  five? 

18.  In  twelve,  how  many  times  three? 

19.  In  twelve',  how  many  times  four? 

20.  In  twenty,  how  many  times  five? 

21.  In  eighteen,  how  many  times  six? 

22.  In  sixteen,  how  many  times  two? 


SIMPLE  DIVISION. 

23.  In  thirty,  how  many  times  five? 

24.  In  thirty,  how  many  times  six? 

25.  In  twenty-one  how  many  times  seven? 

26.  In  twenty-eight, how  many  times  seven? 

27.  In  thirty-six,  how  many  times  twelve? 

28.  In  forty-eight,  how  many  times  twelve? 

29.  In  forty-eight,  how  many  times  sixteen? 

30.  In  fifty-five,  how  many  times  eleven? 

31.  In  sixty  IK;W  many  times  twenty? 
3*2.  In  eighty,  how  many  times  twenty? 

33.  In  one  hundred,  how  many  times  twenty? 

34.  In  one  hundred  and  twenty,  how  many  times  thirty  ? 

35.  In  ten,  how  many  times  four  ? 
Answer.  Two  times,  and  two  remain. 
3f>.  In  fourteen,  how  many  times  three  ? 
Answer.  Four  times,  and  two  remain. 

37.  In  twenty-five,  how  many  times  four? 
Answer.  Six,  and  one  remains. 


There  are  in  Division  four  principle  parts,  viz: 
The  dividend,  or  number  to  be  divided. 
The  dwiwr,  or  number  given  to  divide  by. 
The  quotient,  or  answer,  which  shows  how  mnn 

times  the  divisor  is  contained  in  the  divr-err  . 
The  remainder,  which  is  any  overplus  of  figure, 
that  may  remain  after  the  sum  is  done,  and  i; 
always  less  than  the  divisor. 

CASE  I. 

RULE. — First,  find  how  many  times  the  divisor  if 
contained  in  as  many  figures  on  the  left  hand  of  the 
dividend  as  are  necessary  for  the  operation,  and  place 
the  number  in  the  quotient.  Multiply  the  divisor  l^ 
this  number,,  and  set  the  product  under  the  figures  r' 
the  lef,  hand  of  the  dividend  I  ef  >re  mentioned.  Sul  trac 
this  product  from  that  part  of  the  dividend  under  whicl 
it  stands,  and  to  the  remainder  bring  down  the  nex; 
figure  of  the  dividend;  but  if  this  will  not  contain  th« 
divisor,  place  a  cypher  in  the  quotient,  and  I  rin<r  dcwi< 
another  figure  of  the  dividend,  and  so  on,  until!  it  wi! 


24  SIMPLE  DIVISION. 

contain  the  divisor.  Divide  this  remainder  (thus  in 
creased)  in  the  same  manner  as  before ;  and  proceed  in 
this  manner  until  all  the  figures  in  the  dividend  are 
brought  down  and  used. 


Multiply  the  quotient  by  the  divisor,  and  to  the  pro- 
duct add  the  last  remainder,  if  there  be  any;  if  the 
work  is  right,  the  sum  will  be  equal  to  the  dividend. 

EXAMPLES. 

i, 
Divisor.         Dividend.  Quotient. 

3)143967182(47989060 
12  3 


2  3         Proof    1  43067182 
2  1 

In  this  example,  I  find 
2  9  that  3,  the  divisor,  can 

2  7  not  be  contained  in  the 

first  figure  of  the  divi- 

2  6  dend ;  therefore  I  take 

2  4  two  figures,  v  iz :  14,  and 

inquire  how  often  3  i 

27  contained  therein,which 

27  I  find  to  be  4  times,  and 

put  4  in  the  quotient. — 

1  8  Then  multiplying   the 

1  8  divisor   by  it,  I  set  the: 

.        product  under  the  14, 

Remainder.  2        in   the    dividend,    r.nd 

find  by  subtracting  that 
there  is  a  remainder  of  two.  To  this  2,  I  bring  down 
the  next  figure  in  the  dividend,  viz:  3,  which  increase? 
•he  remainder  to  23.  I  then  seek  how  often  3  is  con- 
tained in  23,  and  proceed  as  before,  When  I  bring  down 
the  one  that  is  in  the  dividend,  I  find  that  three  can  not 
be  contained  in  it,  and  therefore  place  a  cypher  in  the 
quotient  and  bring  down  the  8,  which  makes  18.  Find- 
ing that  3  is  contained  6  times  in  18,  and  that  there  is  no 


SIMPLE  DIVISION.  ^ 

remainder,  I  bring  down  the  2;  but  as  3  can  not  be  con- 
tained in  it,  I  place  a  cypher  in  the  quotient,  and  let  2 
stand  as  the  last  remainder.  In  proving  the  sum  by 
Multiplication,  the  2  is  added.  This  mode  of  operation 
is  called  LONG  DIVISION 

u.  in. 

2)3456789(1788394  5)8789876(1357975 

22  55 


14  Proof  3456789  17  Pr  678987G 

14  15 

5  28 

4  25 

16  39 

16  35 

7  48 

6  45 

18  37 

18  35 

9  26 

8  25 

1  1 

IV.  V. 

42)9870(235         320(12864016081(40200050 
84  1280 

147  640 

126  640 

210  1603 

210  1600 

61 


SIMPLE  DIVISION. 


VI.  VII, 

12  )  301203  (  25100  15  )  218760  (  14584 

24       12  15       15 


61  Pr,301203  68         72920 

60  60        14584 

12  87     218760 

12  75 

-03  126 

120 


60 
60 


VIII. 

848)2468098(3808 
1944 

'3808 

5240  Proof     648 

5184  

- 30464 

5698  15232 

5184  22848 

514 

514 

2468098 

1.  Divide    87654  by    58    Quo.  1511     Rem.     16 

2.  456789         679  672  501 

3.  3875642       7898  490  5622 

4.  98765432       1234  80036  1008 

5.  12486240    87654  142  39372 

6.  57289761       7569  7569 

Note. — When  there  is  one  cypher,  or  more,  at  the 
right  hand  of  the  divisor,  it  may  be  cut  off ;  but  when 
this  is  done,  the  same  number  of  figures  must  be  cut  off 
from  the  right  hand  of  the  dividend ;  and  the  figures 
thus  cut  off,  must  be  placed  at  the  right  hand  of  the  re- 
mainder. 


SIMPLE  DIVISION.  27, 

EXAMPLES. 
I.  II. 

6|00)567434|  10(94572  18|000)246864|593(13714 

54  18 

27  66 

24  54 

34  128 

30  126 


43  26 

42  18 

14  84 

12  72 


210  Remainder         12593 

Note.— In  dividing  by  10,  100,  or  1000,  &c.  when 
you  cut  off  as  many  figures  from  the  dividend  as  there 
are  cyphers  in  the  divisor,  the  sum  is  done ;  for  the 
figures  cut  off  at  the  right  hand  are  the  remainder,  and 
those  at  the  left  are  the  quotient,  as  in  the  following 
sums : 

nr.  iv. 

Quotient.  Quotient. 

1 10)98765(4  Rem.  1 100)123456(78  Rem. 

Quo.  Quo. 

1|000)56789(876  Rem.  1|0000)8765(4321  Rem. 

CASE  II. 

When  the  divisor  does  not  exceed  12,  seek  how  often 
it  is  contained  in  the  first  figure  or  figures  of  the  divi- 
dend, and  place  the  result  in  the  quotient.  Then  mul- 
tiply in  your  mind  the  divisot  by  the  figure  placed  in  the 
quotient,  subtract  the  product  from  the  figure  under 
which  it  would  properly  stand  in  the  former  case  of  di- 
vision and  conceive  the  remainder,  if  there  be  any,  to 
be  prefixed  to  the  next  figure.  See  how  often  the  divi- 
sor is  contained  in  these,  and  proceed,  as  before,  till 
3 


28  SIMPLE   DIVISION. 

;he  whole  is  divided.     This  operation  is  called  SHORT 
DIVISION. 

EXAMPLES. 

I.  II. 

4)  987654321  8)  123456789 

Quo.  24691358  0—1  Quo.  1543209  8—5 
In  the  first  example,  I  find  that  4  is  contained  twice 
in  9,  and  that  1  remains.  The  1, 1  conceive  as  prefixed 
to  the  next  figure,  which  is  8,  and  they  become  18.  In 
18, 1  find  4  is  contained  4  times,  and  2  remain.  By 
prefixing  the  2  to  the  following  figure,  which  is  7,  they 
make  27.  In  this  manner  I  proceed,  setting  the  result 
of  each  calculation  in  the  lower  line  which  is  the  quo 
tient.  In  the  second  example,  as  8  can  not  be  contained 
in  1,  take  two  figures,  and  proceed  as  in  the  first. 

in.  rv. 

9)1023684200  12) 19 14678987 

v.  vi. 

11)6789870062  12)1000001246 

Note. — When  the  divisor  is  of  such  a  number  that  two 
figures  being  multiplied  together  will  produce  it,  divide 
the  dividend  by  one  of  those  figures,  the  quotient  thenc< 
I  irising,  by  the  other  figure,  and  it  will  give  the  quotien 
required.  As  it  sometimes  happens  that  there  is  a  re 
tnainder  to  each  of  the  quotients,  and  neither  of  then 
I  "he  true  one,  it  may  be  found  thus : — Multiply  the  firs 
divisor  by  the  last  remainder,  and  to  the  product  adc 
the  first  remainder,  which  will  give  the  true  one. 

EXAMPLES. 
I. 

Divide  249738  by  56. 
8]  249738 

8 

7 | 312  17— 2  4 


4459--*  32 

2 


34  Remainder. 


SIMPLE    DIVISION. 


29 


The  same  done  by  Long  Division. 
561249738F4459 
224 


333 

280 


538 
504 

34  Remainder. 

ii. 

Divide  1847562324  by  84. 
12] 1847562324         7] 1 847562324 

7]153963527       12] 26393747  4—6 


2199478  9—4 
12 


4  8  Rem, 


2199478  9—6 

7 

-    42 
6 

Rem.  48 


3.  Divides  463  098  6    by        72. 

4.  "  6  7  8  6  0  1  2  1     by         63. 

5.  «  124  5  6743     by         96. 

6.  «          3  4  2  1  0  3  9  0    by         81. 

7.  «          5  4  6  9  7  2  8  3     by      103. 

8.  «  7  5  3  9  2  6  1  8     by      113. 

Note. — In  all  cases  in  Division,  when  there  is  any  re- 
mainder, the  remainder  and  divisor  form  a  Vulgar 
Fraction.  Thus,  if  the  divisor  be  8  and  the  remainder 
5,  they  make  f  or  five  eights ;  or,  as  in  one  of  the  pre- 
ceding examples,  the  divisor  is  56  and  the  remainder  34, 
which  make  . 


30  SIMPLE  DIVISION, 

APPLICATION. 

1.  A  man  bought  6  oxen   for  318  dollars.     How 
much  did  he  pay  a  head.  Ans.  53  dollars. 

2.  How  much  flour  at  7  dollars  per  barrel  can  be 
bought  for  1512  dollars?  Ans.  216  barrels, 

3.  If  1600  bushels  of  corn  are  to  be  divided  equally 
among  40  men,  how  much  is  that  a  piece? 

Ans.  40  bushels. 

4.  The  salary  of  the  President  of  the  United  States 
is  25000  dollars  a  year.     How  much  is  that  a  day, 
reckoning  365  days  to  the  year?  Ans.  68-iff 

5.  A  regiment  consisting  of  5CO  men  are  allowed 
1000  pounds  of  pork  per  day.     How  much  is  each 
man's  part?  Ans.  2  pounds. 

6.  If  a  field  of  32  acres  produce   1920  bushels  of 
corn,  how  much  is  that  per  acre?          Ans.  60  bushels. 

7.  A  prize  of  25526  dollars  is  to  be  equally  divided 
among  100  men.     What  will  be  each  man's  part? 

Ans.  255TW  dollars. 

8.  How  many  oxen  at  30  dollars  a  head,  may  be 
bought  for  38040  dollars?  Ans.  1268  oxen. 

Question  1.  What  does  Simple  Division  teach? 

2.  What  are  the  four  principal  parts  of  Di 

vision? 

3.  How  do  you  proceed  when  there  is  one  cy 

pher  or  more  on  the  right  hand  of  the 
divisor? 

4.  How  do  you  proceed  in  dividing  by  ten,  or 

a  hundred,  or  a  thousand? 

5.  How  do  you  proceed  when  the  divisor  does 

not  exceed  12? 

6.  When  you  divide  by  nny  number  not  ex- 

ceeding 12,  what  is  the  operation  called? 

7.  When  the  divisor  is  of  such  a  number  that 

two  figures  multiplied  together  will  pro- 
duce it? 

8.  What  can  be  made  by   placing  the  remain- 

der of  a  sum  over  the  divisor?     Ans.  A 
Vulgar  Fraction. 

9.  How  is  a  sum  in  Division  proved? 


moniscrors  QUESTIONS.  31 

Promiscuous  Exercises  on  the  slate  under  all  the  fore- 
going Rules. 

1.  A  man  bought  a  cart  for  25  dollars,  a  yoke  of  oxen 
for  69  dollars,  and  a   plough  for  7  dollars.     What  did 
Sie  give  for  the  whole?  Ans.  101  dollars. 

2.  What  will  315  bushels  of  rye  cost  at  42  cents  a 
ushel?  Ans.  13230  cents. 

3.  If  my  income  be  1647  dollars,  and  I  spend  1010 
dollars  of  it,  how  much  do  I  save?      Ans.  637  dolla.rs. 

4.  Four  boys  had  gathered  113  bushels  of  walnuts; 
in  dividing  them  equally,  how  many  will  each  have? 

Ans.  284  bushels. 

5.  There  are  63  gallons  in  a  hogshead.     How  mapy 
gallons  are  there  in  25  hogsheads?     Ans.  1575  gallons. 

6.  A  merchant  bought  a  stock  of  goods  for  30250  dol- 
lars, and  sold  it  again  for  40000  dollars.     How   much 
did  he  gain?  Ans.  9750  dollars. 

7.  A  merchant  bought  at  auction,  broadcloth  for  350 
dollars,  muslin  97  dollars,  linen  1010  dollars,  silk  874 
dollars,  and  calico  8  dollars.     To  what  did  the  whole 
amount?  Ans.  2339  dollars. 

8.  There  are  328  rows  of  corn  in  my  field  and  each 
row  has  169  hills.     How  many  hills  are  there  in  the 
field?  Ans.  55432  hills 

9.  John  had  in  his  desk  1000  dollars.     lie  took   cut 
120  dollars  to  pay  a  debt;  he  afterwards  put  in  75  dol- 
lars.    How  much  was  there  in  the  desk? 

Ans.  955  dollars. 

10.  A  farmer  has  a  flock  of  60  sheep;  one  third  of 
them  are  black  and  the  rest  white.     How  many  of  them 
[are  black?  Ans.  20. 

11.  A  merchant  has  50  boxes  of  raisins  with   17 
pounds  in  each  box.     How  many  pounds  are  there  in 
all?  Ans.  850  pounds. 

12.  There  are  4  quarts  in  a  gallon.     How  many  gal 
Ions  are  there  in  760  quarts?    ..  Ans.  190  gallons. 

13.  Ann  had  a  paper  of  pins  which  had  600  in  it 
when  she  bought  it,  but  she  used  245  of  them.     How 
many  are  left?  Ans.  355  pins. 

14.  If  I  divide  364  cents  among  14  boys,  how  many 
will  each  have?  Ans.  26  cents 


32  PROMISCUOUS    QUESTIONS. 

15.  There  are  1681  nuts  in  a  basket.     James  took 
out  150,  Charles  272  and  John  1005;   after  which,  Jo- 
seph put  in  95.     How  many  were  there  in  the  basket? 

Ans.  349  nuts. 

16.  A  cooper  worked  115  days,  and  made  6  barrels 
each  day.     How  many  barrels  did  he  make? 

Ans.  690  barrels. 

17.  A  person  has   in  money   5000   dollars;  in  bank 
stock  3500  dollars,  and  in  merchandize  12500  dollars, 
i He  intends  to  divide  all  this  property  equally  among  his 
13  sons.     What  will  be  the  share  of  each  son? 

An?.  7000  dollars. 

18.  William  had  372  pears;  he   kept  120  of  them, 
and  divided  the  rest  between  his  two  sisters.     How  ma- 
ny did  each  sister  receive?  Ans.  126  pears 

ID.  There  are  10  bags  of  coffee  weighing  each  120 
pounds,  and  12  bags  weighing  each  135  pounds.  What 
is  the  weight  of  the  whole?  Ans.  2820  pounds. 

20.  There  are  15  firkins  of  butter  each  weighing 
49  pounds.     The  fh'kins  which  contain  the  butter  wei/ 
each  7  pounds.     How  much  would   the  butter  weigh 
without  the  firkins?  Ans.  630  pounds 

21.  A  man  died,  leaving  12426  dollars  in  cash.     He 
directed  in  his  will  that  1000  dollars  should  be  given  to 
his  niece;  and   that  the  remainder   should  be  equally 
divided  between  his  two  nephews?     What  is  the  share 
of  each  nephew?  Ans.  5713  dollars 

22.  A  farmer,  who  had  a  farm  of  520  acres,  bought 
an  adjoining  one  of  375  acres,  and  divided  the  whole 
equally  among  his  5  sons.     How  many  acres  had  each 
son?  Ans.   179  acres 

23.  A  merchant  bought  5  pieces  of  linen  containing 
25  yards  each,  and  2  pieces  containing  24  yards  each 
and  1  piece  containing  26  yards.     How  many  yards 
were  there  in  the  whole?  Ans.  199  yards, 

24.  Three  boys   bought  3  baskets,  each   contain- 
ing 150  apples,  and  2  barrels,  each  containing  540 
apples.     They  found  219  to  be  rotten,  which  they  threw 
nway  and  divided  the  rest  equally  among  themselves, 
How  many  had  each  for  his  share?      Ans.  437  apples 


FEDERAL  MONEY. 

The  denominations  of  Federal  money,  or  the  money 
)f  the  UNITED  STATES,  are,  Eagle,  Dollar,  Dime,  Cent, 
lid  Mill. 

TABLE. 

10  Mills  (m)  make  1  Cent,  c. 

10  Cents     -  -       1  Dime,  d. 

10  Dimes    -  -       1  Dollar,  /?,  or 

10  Dollars  -  -       1  Eagle,  E. 

In  writing  Federal  Money,  it  is  customary  to  omit 
Uagles,  Dimes,  and  Mills,  and  set  down  sums  in  dol- 
ars,  cents,  and  parts  of  a  cent.  The  parts  of  a  cent 
generally  used  are,  halves,  thirds,  and  quarters.  Thus, 

is  a  half;  *  a  third;  \  a  quarter. 


Exercises  for  tfic  learner. 

1.  How  many  mills  make  a  cent? — How  many  half  a 
cent? — How  many  a  cent  and  a  half? — How  many  two 
cents? 

2.  How  many  halves  of  a  cent  make  a  cent? 

3.  How  many  thirds  of  a  cent  make  a  cent? 

4.  How  many  fourths  of  a  cent  make  a  half  cent? 

5.  How  many  fourths  make  a  cent? 

6.  How  many  cents  make  one  fourth  or  quarter  of  a 

dollar. 

7.  How  many  cents  make  a  half  dollar? 

8.  How  many  cents  make  three-fourths  of  a  dollar? 

9.  How  many  cents  make  a  dollar? 

10.  How  many  dollars  and  cents  in  one  hundred  and 
ten  cents? — How  many  in  two  hundred  and  six  cents?— 
How  many  in  three  hundred  and  forty-eight  cents? — 
How  many  in  five  hundred  and  one  cents? 

11.  If  you  give  a  dollar  for  a  book;  thirty  cents  for  a 
slute,  and  one  cent  for  a  pencil,  how  many  cents  will 
you  give  for  the  whole? 

12.  Write  down  one  dollar  arid  eight  cents.    TWO  dol- 
lars and  sixteen  cents.     Twenty  dollars  and  five  cents? 

13.  Write  down  three  hundred  dollars  and  forty  cents 

14.  Five  hundred  eighty-four  dollars  and  fifty  cents. 


4 


O*  FEDERAL    MONEY. 

15.  Eight  hundred  sixty  dollars  and  sixty-seven  cents. 

16.  Four  thousand  eight  hundred  dollars  and  two  cents. 

17.  Six  hundred  thirty-one  dollars  fifty-six  and  a  fourth 
cents. 

18.  Nine  hundred  and  eighty-seven  dollars. 

19.  Thirty -two  thousand  five  hundred  dollars  eighty 
seven  and  a  half  cents. 

20.  Ten  dollars  sixty-eight  and  three-fourth  cents. 

21.  Twelve  dollars  ninety-three  and  three-fourth  cents. 

22.  Twenty  dollars  thirty-seven  and  a  half  cents. 

23.  Thirty-three  dollars  thirty-three  and  a  third  cents. 

24.  Sixty  dollars  sixty -six  and  two  third  cents. 

25.  Read  the  following  sums,  viz. 

$3448.87*    $3450.25  $47967.91   $7.10  $115.334 
$170.931        $19.01       $85.06^ 


ADDITION   OF  FEDERAL  MONEY. 

RULE. 

Begin  at  the  right  hand  side  of  the  sum,  add  one  row 
of  figures  at  a  time,  and  carry  one  for  every  ten,  from 
the  lower  denomination  to  the  next  higher,  as  in  Simple 
Addition,  until  the  whole  is  added.  When  you  come  to 
the  hist  row  on  the  left  hand,  instead  of  setting  down 
what  remains  over  ten,  twenty,  or  thirty,  &c.  set  down 
the  fall  amount. 

Note. — When  there  are  parts  of  a  cent  in  a  sum,  such 
is  halves,  &c.  find  the  amount  of  them  in  fourths  of  a 
lent;  consider  how  many  cents  these  fourths  will  make, 
ind  add  them  to  the  first  row  in  the  column  of  cents. — 
When  the  parts  of  a  cent  are  not  sufficient  to  make  a 
^ent,  place  their  amount  at  the  right  hand  of  the  column 
-jf  cents,  as  in  the  first  example ;  and  when  the  parts  of 
i  cent  make  one  cent  or  more,  and  some  parts  remain, 
but  not  enough  for  another  cent,  the  parts  thus  remaining 
mast  be  set  down  in  the  same  way,  according  to  the  se- 
cond example.  The  proof  is  the  same  as  in  Simple  Ad- 
dition. 


FEDERAL  MONEY. 


35'' 


EXAMPLES. 


I. 

D.  cts. 
5432.12* 
1234.564 

7898.76 
5432.12 
3456.78 


ii. 

D.  cts. 
324.87* 
987.431 
720.30 
842.431 
103.62* 


in. 

D.  cts. 
885.90 
125.87* 
440.40 
867.12i 
390.97 


IV, 

D.  cts. 
987654.32 
123456.78 
£87654.32 
123000.45 
678987.65 


23454.341   2975.67*   2710.27   2900753.52 


APPLICATION. 

1.  A  man  bought  a  farm  in  five  parcels;  for  the  first, 
he  gave  $250.75;  for  the  second,   $350;  for  the  third, 
$475,87*  :  for  the  fourth,  $550;  and  for  the  fifth,  $600. 
What  was  paid  for  the  farm?  Ans.  2226.62* 

2.  A  merchant,  in  buying  gave  for  flour,  $325.43! ; 
for  sugar,  $854.25;  for  molasses,  $520.62* ;  for  coffee, 
$944.50 ;  and  for  cotton,  $6427.12*.     What  was  the  sum 
paid?  Ans.  $9071.931. 

3.  What  is  the  amount  of  10*  cents;  93!  cents;  87* 
cents;  50  cts.;  314  cts.:  43!  cts.;  and  11  dollars? 

Ans.  $14.16!  cents. 

4.  Gave  for  an  Arithmetick  314  cents;  for  asiate,  37* 
cents;  for  quills,  50  cents;  for  an  inkstand,  62*  cents; 
for  a  Geography,  1  dollar,  and  for  a  History,  87*  cents. 
How  much  do  they  amount  to?  Ans.  $3.68!  cents 

5.  Add    $75212.50,      $544225.75,      $4587220.50, 
$90000,  and  $5876432.75. 


SUBTRACTION  OF  FEDERAL  MONEY. 


RULE. 


Place  the  smaller  sum  under  the  larger,  setting  the 
dollars  under  dollars  and  cents  under  cents, and  proceed 
as  in  Simple  Substraction.  When  there  is  a  fraction,  or 
part  of  a  cent  in  the  upper  line  of  figures,  and  none  in 
the  lower,  set  it  down  at  the  right  of  the  remainder,  a.-- 


36  FEDERAL    MONEY. 

a  part  of  the  answer.  When  there  is  a  fraction  in  each 
line,  and  the  upper  one  is  the  larger,  subtract  the  lower 
one  from  it  and  set  clown  the  difference ;  but  if  the  lower 
one  is  larger  than  the  upper,  subtract  it  from  the  num- 
ber that  it  takes  of  the  fraction  to  make  a  cent — add 
the  difference  to  the  upper  one,  and  set  down  the  amount. 
When  there  is  a  fraction  in  the  lower  line  and  none  in 
the  upper;  subtract  the  fraction  irom  the  number  that 
it  takes  of  it  to  make  a  cent,  and  sat  down  the  remain- 
der. In  this  case,  and  likewise  when  the  part  or  frac- 
tion I  clow  is  larger  than  the  upper  one,  it  is  necessary 
to  cany  one  to  the  right  hand  figure  of  the  lower  row 
of  cente^ 

KXAXPLF.3. 

IT.  ni.        iv. 

7).  c.  D.  c.       D.  c. 

587.25  687.31  9000.43 

292.50  500.81  8220.314 


$277.15  $294.75  $37.50     $780.111 

v.  vr.  vn.  vin. 

I),  c.  D.  c.  D.  c.  J).     r. 

6U5.624  820.431  5078.314  9810000.12! 

457.87*  790.37*  4689.031  1037654.68* 


$237.75  $30.064         $1288.37*  $7822345.44-! 

9.  Subtract  $ 387.20  from  glOOO. 

10.  Subtract  $'5871.31:1,  from  $5430.87*. 

11.  Take  $44.874  from  300  dollars. 

12.  Take  $11000,  from  $19876.87*. 

APPLICATION. 

1.  Bought  goods  amounting  to  $3875.62*,  and  hav- 
ing paid  $2350.93*;  how  much  remains  due? 

Aw.  2024.681. 

2.  My  account  .ipiirtst  my  neighbour  amounts  to  $759. 
25;  and  his  account  against  me  is  $346.87*.      How 

| much  does  he  owe  me? "  ADO.  212.37*. 


FEDERAL    MONEY.  37 

3.  Having  bought  a  quantity  of  goods  at  $5425,  and 
sold  them  at  $3932.681.     How  much  did  I  make  on  the 
goods?  Ans.  $1507.6^. 

4.  A  owes  me  $11587.50,  but  having  failed  in  busi- 
ness, he  is  able  to  pay  $9263.62 L     How  much  do  I  lose? 

Ans.  $2323.874 

5.  Subtract  $3427.874,  from  $9000.    Ans.  $572.124. 


MULTIPLICATION  OF  FEDERAL  MONEY. 

BULE. 

Set  the  multiplier  under  the  sum,  and  proceed  as  in 
Simple  Multiplication,  carrying  one  for  every  ten  from 
a  lower  to  a  higher  denomination,  until  the  whole  is 
multiplied.  After  the  sum  is  done,  separate,  by  a  pe- 
riod, the  two  right  hand  figures  of  the  product  for  cents, 
and  the  figures  at  the  left  hand  of  the  period  will  be 
dollars. 

Note. — When  the  sum  to  be  multiplied  contains  a  frac- 
tion, or  part  of  a  cent,  multiply  it  by  the  multipler,  and 
consider  how  many  cents  are  contained  in  its  product. 
Then  multiply  the  first  figure  of  the  cents  and  add  to 
its  product  the  cents  contained  in  the  product  of  the 
fraction,  and  proceed  as  directed  above.  In  multiplying 
a  fraction,  if  you  find  in  the  product  one  cent  or  more, 
and  a  remainder  not  large  enough  to  make  another  cent, 
set  down  the  fraction  at.  the  right  hand  of  the  product, 
that  is  under  the  row  of  fractions  or  parts  of  a  cent. — 
When  there  is  a  fraction  in  the  sum,  and  the  multiplier 
exceeds  12,  multiply  the  sum  without  the  fraction,  and 
afterwards  multiply  the  fraction  and  add  it  to  the  sum. 

EXAMPLES. 
I.  II.  III.  IV.  V. 

D.  c.         D.  c.          D.  £.         D,  c.          D.  c. 
124.10       830.12*       172.30      2451.624     275.431 
23  4  5  12 


248.20    2490.374      689.20      12258.12*     3305.25 


FEDERAL    MONEY. 

VII. 
D.      C. 

3120.17 
24 


1314600.88 
1643251.1 

2957851.98 


12480.68 
62403.4 

74884.08 


9. 
10. 
11. 
12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 


Multiply 


$420.50 
519.75 
99.62* 
75.314 
62.12* 
750.25 
330.12* 
248.87* 
95.931 
24.17 
37.50 
58.931 
9876.624 


by 

by 

by 
by 


3 

4 
5 

by   6 

7 


Ans. 
Ans. 
Ans. 
Ans. 
Ans. 
Ans. 
Ans. 
Ans. 
Ans. 
Ans. 
Ans. 
Ans. 


by 

by  8 

by  9 

by  12 

by  10 

by  28 

by  36 

by  208  Ans.  2054338.00 


1542.17 
6609.3 
94 

8151.564 

$841.00 

1559.25 

398.50 

376.564 

372.75 

5251.75 

2641.00 

2239.87* 

1151.25 

241.70* 

1050.00 

2121.75 


APPLICATION. 

1.  How  much  will  18  barrels  of  flour  cost,  at  3  dol- 
lars per  barrel  ?  Ans.  54  dollars. 

2.  What  will  35  pounds  of  coffee  cost,  at  20  cents  per 
pound?  Ans.  7  dollars. 

3.  Sold  87  barrels  of  flour,  at   $3.12*    per  barrel. 
What  was  the  amount?  Ans.  $271.87*. 

4.  Bought  160  acres  of  land,  at  $1.25  per  acre. — 
What  did  the  whole  cost?  Ans.  200  dollars. 

5.  What  will-  225  bushels  of  apples  cost,  at  62*  cents 
per  bushel?  Ans.  140.62*. 

6.  What  will  5SO  bushels  of  salt  cost,  at  $1.12*  per 
btishe  ?  Ans.  $652.50, 

*Io  multiplying  by  10,  when  there  is  no  fraction  in  the  sum,  it  is 
necessary  to  add  a  cypher  to  the  rierht  hand  of  the  sum,  placing  the 
period  that  separates  cents  from  dollars  one  iigure  farther  towaros  the 
rlerht  hand,  anrl  the  sum  is  done.  In  multiplying  by  100t  add  two  cy- 
phers: by  1000,  thret:-,  &c. 


FEDERAL    BS.OKEY.  3J 

DIVISION  OF  FEDERAL  MONEY. 

RULE. 

Proceed  as  in  Simple  Division.  When  the  sum  con- 
sists of  dollars  and  cents,  the  two  right  hand  figures  of 
the  quotient  will  be  cents.  When  there  is  a  remainder, 
multiply  it  by  4,  adding  the  number  of  fourths  that  are 
in  the  fraction  of  the  sum  (if  there  be  any)  to  its  pro- 
duct; then  divide  this  product  by  the  divisor,  nnd  its 
quotient  will  be  fourths3,  which  must  be  annexed  to  the 
quotient  of  the  sum.  When  the  sum  consists  of  dollars 
only,  if  there  he  a  remainder,  add  two  cyphers  to  it; 
then  divide  by  the  divisor  as  before,  and  its  quotient 
will  be  cents,  which  must  be  added  to  the  quotient  of 
the  sum.  When  the  sum  is  in  dollars,  and  the  divisor  is 
larger  than  the  dividend,  add  two  cyphers  to  the  divi- 
dend— then  divide,  and  the  quotient  will  be  in  cents. 
• 

EXAMPLES. 
I.  II.  III.  IV. 

D.  c.          D.      c.  D.  c.  D.  c.  D.  c. 

2)120.50       4)SOOO.OO         5)580.75      7)34.4C(12.07 


210.25          2000.00  HG.io          — 

14 
14 


49 

49 


T.                                       VI.  VII. 

D.  c.              D.  c.    D.  c.  D.  c.     DC 

£(27.81(3.09      12)144.60(12,05  38)162.36(4.51 

27                        12  144 

81                        24 .  183 

81                        24  180 

60  36 

60  36 


40 

FEDERAL    MONEY* 

VIII. 

IX. 

D.    c.    D. 

c. 

D.    c.      D.  c. 

35)1234.72(34.291 

44)87654.32(1992.144 

108 

44 

154 

436 

144 

396 

107 

405 

72 

396 

352 

94 

324 

88 

28 

63 

4 

44 

36)112(3 

192 

108 

176 

4 

16 

—  . 

' 

4 

44)64(1 

44 

20+ 

10.     Divide 

$840 

by            12 

11.         « 

717.12J 

by              8 

12.         " 

246.25 

by              9 

13.        « 

687  .  20 

by             12 

14.         « 

980 

by            34 

15.         « 

87654 

by           128 

16.         « 

1284 

by          112 

17.         « 

40000 

by           188 

18.         « 

976  .  87* 

by          225 

19.        " 

1334.37* 

by           212 

20.        « 

9876  .  44 

bv          345 

21.        " 

89786.54 

by           374 

FEDERAL   MONEY.  41 

APPLICATION. 

1.  Divide  400  dollars,  equally,  among  20  persons. 
What  will  be  the  portion  of  each  person  ?      Ans.  $20. 

2.  Divide  1728  dollars,  equally  among  12  persons. 
What  does  each  one  of  them  share?  Ans.  $144. 

3.  If  240  bushels  cost  420  dollars;  what  Is  the  cost  of 
one  bushel  at  the  sams  rate?  Ans.  $1.75. 

Promiscuous  Examples. 

1.  What  will  the  following  sums  amount  to,  when  ad- 
ded together,  namely  r — 

$124.62* ;       $248.874 ;       $342.40 ;       $9850.25. 
and  $20.314?  Ans.  $10588.464. 

2.  If  my  estate  is  worth  12870  dollars,  and  1  meet 
with  losses  amounting  to  $4364.50,  how  much  shall  J 
have  left?  Ans.  $8505.50. 

3.  A  merchant  enters  into  a  trade  by  which  he  re- 
ceives $1324.624  per  yeai\  for  four  years:  how  much 
is  his  whole  gain?  Ans.  $5298.50. 

4.  An  estate  of  98740  dollars  is  to  be  divided, equally, 
between  8  heirs;  what  will  each  receive? 

Ans.  $12342.50. 

A  bought  of  B, 

1  barrel  of  sugar  at  -  -  $24.50 

1  chest  of  tea,          -  -  -  60.00 

1  hogshead  of  salt,  -  -  3.75 

20  yards  of  cloth,      -  -  -  15.00 

1  barrel  of  flour,      -  3.87  i 


Ans.  $107.124. 


Q.  1.    What  are  the  denominations  of  Federal  Money  ? 

2.  How  many  mills  make  a  cent? 

3.  How  many  cents  make  a  dime? 

4.  How  many  dimes  make  a  dollar? 

5.  How  many  dollars  make  an  eagle? 

6.  How  are  the  denominations  generally  used   in 

writing  Federal  Money,  and  in  reckoning? 

7.  Where  is  Federal  Money  used  as  a  currency? 
Answer.     In  the  United  States  of  North  America. 

4 


42 


FEDERAL  MOXEY. 


Exercises  for  the  slate  in  Federal  Money. 

1.  What  is  the  sum  of  50  cents  and  5  dimes? 

Ans.  1  dollar 

2.  A  man  owed   $35.46.5,  and  paid  $27.69.6;  how 
[much  did  he  th.on  owe?  Ans.  $8.76.9 

3.  A  man   laboured  6  months  at  25  dollars  6  cents 
and  5   mills  per  month.     How  mr.ch  did  his   wa 
amount  to?  Ans.  $150.3$ 

4.  If  you  divide  $35001.50  equally  among  125  men 
how  many  dollars  will  each  have?        Ans.  $280.01,2 

5.  Bought  an   umbrella  for  four  dollars,  a  penknife 
for  37  cejts  and  5  mills,  a   hat  for  3  dollars  and  6* 
cents,  a  cane  for  one  dollar  and   5  dimes,  and  a  book 
for  one  dollar  and  a  quarter.     How  much   did  he   p 
for  all  these  articles.  Ans.  $10.19 

6.  What   will   1473  peaches  come  to  at  5   mills  u 
piece?         Ans.  7  dollars  36  cents  5  mills,  or  $7.38.5 

7.  A  lady  bought  8  yards  of  silk  for  10.37.6.     How 
much  is  that  a  yard?  Ans.  $1.29.7 

8.  A  boy  sold  a  top  for  a  dollar,  and  gave  away  12 2 
cents  of  the  money.     How  much  had  he  left? 

Ans.  87i  cents 

9.  Add  two  mills,  two  cents  two  dimes,  and  two  dol- 
ars  together.  Ans.  $2.22.2 

19.  You  borrow    $535.15  and   pay  $236.18;  how 
much  remains  unpaid.  Ans.  299.97. 

11.  Going  a  journey  I  took  two  50  dollar  bills  to 
oear  my  expenses,  which  were  as  follows,  viz:  Stage 
'are  eighteen  dollars;  board,  nine  dollars  and  fifty  cents; 
carrying    trunk,    seventy-five  cent$;  private  convey- 
ance at  one  time,  six  dollars  and  37  i  cents;  and  at  an- 
ather,  seven  dollars.     How  much  had  I  left,  on  my  re- 
urn  home?  Ans.  58^37* . 

12.  From  five  dollars  take  one  mill?     Ans.  $4.99.9. 

13.  From  4  dollars  take  3  dollars  99  cents  and  9 
mills.  Ans.  $0.00.1. 

14.  What  will  2640  bushels  of  oats  cost  at  12*  cents 
[>er  bushel?  Ans.  $330. 

15.  A  man  paid  $£fr.60  for  12  yards  of  cloth.    How 
much  was  the  cloth  a  yard?  Ans.  $2.30. 


FEDERAL  MONEY.  43 

16.  If  John  spends  6  cents  a  day,  how  many  would 
he  spend  in  a  year,  or  365  days?  Ans.  $21.90. 

17.  I  bought  a  wagon  for  70  dollars,  and  paid  $9.124 
for  repairing  ir,  and  sold  it  again   for  $75.624.     Did  I 
make  or  lose  by  the  bargain?  and  how  much? 

Ans.  Lost  $3.50. 

18.  I  bought  at  a  store  a  pair  of  gloves  for  75  cents, 
a  vest  pattern  for  $1.12* ,  two  yards  of  cloth  for  a  coat 
lat  $2.874  per  yard,  and  trimmings  for  18!  cents;  and 
jto  pay  the  bill,  I  gave  the  merchant   an  eagle.     How 
much  should  he  give  me  in  change?          Ans.  $2.18!. 

19.  Three  men  sold  their  marketing  for  10   dollars. 
What  would  be  the  share  of  each,  if  the  money  were 
divided  equally  among  them?  $3.334 

20.  A  lady  went  a  shopping  with  the  sum  of  55  dol- 
lars and  her  purchases  amounted  to  29  dollars  and  624 
|cents;  how  much  had  she  remaining?      Ans.  $25.374. 

21.  If   a  person   pay  for  board  $3.50   a  week,  how 
much  will  he  pay  in  26  weeks?  Ans.  $91.00. 

22.  A  person  bought  a  box  of  tea  weighing  14  pounds 
for  $17.50;  how  much  is  it  a  pound?  Ans.  $1,25. 

88.  A  gentleman  rents  a  house  for  650  dollars  a 
year;  how  much  is  it  a  month?  Ans.  $54.16.6. 

24.  A  person  has  due  to  him  the  following  sums  of 
money,  viz:  574  dollars  64  cents;  425  dollars  25  cents; 
175  dollars  68!  cents;  341  dollars  874  cents;  1021  dol- 
lars 124  cents.     How  muclxis  due  to  him  in  all? 

Ans.  $2538.00. 

25.  Bought  35  yards  of  cloth  at  2  dollars  and  64 
cents  a  yard;  51  pair  of  shoes   at  874  cents  a  yard;  10 
'dozen  of  buttons,  at  16!  cents  a  dozen,  and  75  hats,  at 
$3.25  a  piece.     To  how  much  did  the  whole  amount? 

Ans.  362.43!. 

28.  Four  men  about  to  descend  the  Ohio  river  in  a 
boat,  laid  in  the  following  provisions,  viz:  26  loaves  of 
bread,  at  64  cents  a  loaf:  41  pounds  of  ham,  at  124 
cents  a  pound ;  17  pounds  of  coffee  at  18!  cents  a 
pound,  15  pounds  of  cheese  at  9  cents  a  pound,  and  paid 
for  sundry  other  articles  $6.434.  How  much  did  the 
whole  amount  to,  and  what  was  the  share  of  each? 

Ans.  $17.72.     Each  man's  share  $4.43. 


TABLE 

OF 

MONEY,  WEIGHTS,  MEASURES,  &c^ 

ENGLISH  MONEY. 
A  table  of  Federal  Money  has  already  been  given. 

The  denominations  of  English  Money  are  pound,  shil- 

ling, penny,  and  farthing. 

4  farthings  (qr.)         make         1  penny 

d. 

12  pence  -                                      1  shilling 

s. 

20  shillings        -                            1  pound 

£• 

OirFarthings  are  written  .as  fractions,  thus  : 

4  one  farthing. 

h  two  farthings,  or  a  half-psnny. 

1  three  farthings. 

PENCE    TABLE.                                SHILLING    TABLE. 

d.                             s.     d. 

s. 

£•  *» 

20  pence  make           1     8 

20  shillings  make 

1     0 

30      "         "-26 

30 

1  10 

40      «         "-34 

40 

2    0 

50      «         "-42 

50 

2  10 

60      "         "-50 

60 

3    0 

70      "         "      -      5  10 

70 

3  10 

80      "         "-68 

80 

4    0 

90      "        "-76 

90 

4  10 

100      "         "-84 

100 

5     0 

110      "        "-92 

110 

5  10 

120      «         "      -     10     0 

120 

6    0 

240      «        «      -    20    0 

130 

6  10 

TABLE   CP    WEIGHTS    AND    MEASURES.  45 

TROY  WEIGHT. 

By  this  weight,  jewels,  gold,  silver,  and  liquors  are 
weighed. 

The  denominations  of  Troy  Weight  are  pound, 
ounce,  pennyweight,  and  grain. 

24  grains  (gr.)         make         1  pennyweight  dwt. 

20  pennyweights  1  ounce  ox. 

12  ounces       -  1  pound  lb, 

AVOIRDUPOIS  WEIGHT. 

By  this  weight  are  weighed  things  of  a  coarse  dros- 
sy nature,  that  are  bought  and  sold  by  weight;  and  all 
metals  but  silver  and  gold. 

The  denominations  of  Avoirdupois  Weight  are  ton, 
hundred  weight,  quarter,  pound,  ounce,  and  dram. 

16  drams,  (dr.)      make      1  ounce     -  -     ox. 

18  ounces      -  1  pound     -  -     lb. 

28  pounds     -  1  quarter  of  a  cwt.    qr. 

4  quarters,  or  112  lb.        1  hundred  weight     cwt. 

20  hundred  weight.  1  ton.        -  -      T. 

APOTHECARIES  WEIGHT. 

By  this  weight  apothecaries  mix  their  medicines,  but 
buy  and  sell  by  Avoirdupois  Weight. 

The  denominations  of  Apothecaries  Weight  are  pound, 
ounce,  dram,  scruple,  and  grain. 

20  grains  (gr.)          make  1  scruple         9 

3  scruples  -  1  dram  3 

8  drams      -  -  -       1  ounce          3 

12  ounces     -  1  pound          fc 

LONG   MEASURE. 

Long  measure  is  used  for  lengths  and  distances. 
The  denominations   of  Long    Measure   are  degree, 
league,  mile,  furlong,  pole,  yard,  foot,  and  inch. 


45  TABLE    CF    MEASURES. 

12  inches  (in.)     make  1  foot  -     ft. 

3  feet  1  yard    -  -     yd. 

5i  yards,  or  10£   foot  1  rod,  pole, or  perch  P. 

40  poles  (or  220  yds.)  1  furlong  -         -    fur. 

8  furlongs  (or  1760  yds.)  1  mile  -         -     M . 

3  miles  1  league  -              L. 

60  geographick      |       „  l    ,  ,         _      . 
or  6fii  statute  \ 

Note. — A  hand  is  a  measure  of  4  inches,  and  used  in 
measuring  the  height  of  horses. 

A  fathom  is  6  feet,  and  used  chiefly  in  measuring  the 
depth  of  water. 

CUBICK,  OR  SOLID  MEASURE. 

By  Cubick,  or  Solid  Measure,  are  measured  all  things 
that  have  length,  treadth  and  thickness. 

Its  denominations  are,  inches,  feet,  ton,  or  load,  and 
cord. 

1728  inches  make        -         1  cubick  foot. 

27  feet   -  -  -1  yard. 

40  feet  of  round  timber) 

or  50  feet  of  hewn>  -         1  ton  or  load, 

timber.  ) 

128  solid  feet,  i.  e.  8  in) 

length,  4  in  breadth,)  1  cord  of  wood 

and  4  in  height.      ) 

LAND,  OR  SQUARE  MEASURE. 

This  measure  shows  the  quantity  of  lands. 
The  denominations  of  land  Measure  are  acre,  rood, 
square  perch,  square  yard,  and  square  foot. 

144  square  inches         make         1  square  foot    ft. 

9  square  feet  •  1  square  yard  yd. 

304  square  yards          -  1  square  perch  P. 

40  square  perches  1  rood       -        J?. 

4  roods  1  acre       -        A. 

640  acres  1  mile       -        m. 


TABLE    OF     MEASURES.  47 

CLOTH  MEASURE. 

By  this  measure  cloth,  tapes,  &,c.  are  measured. 
The  denominations  of  Cloth  Measure  are  English  ell, 
Flemish  ell,  yard,  quarter  of  a  yard,  and  nail. 
4  nails  (na.)         make         1  quarter  of  a  yard  qr, 

4  quarters  1  yard  -    yd. 

3  quarters  •         -         1  ell  Flemish      -  E.F1. 

5  quarters  -         -         1  ell  English       -    E.  E. 

6  quarters  -        -         1  ell  French      -    E.F. 

DRY  MEASURE. 

This  measure  is  used  for  grain,  fruit,  salt,  &c. 
The  denominations  of  Dry  Measure  are  bushel,  peck, 
quart,  and  pint. 

2  pints  (jpt.)         make         1  quart     -         -         qt. 
8  quarts     -  1  peck      -  pe. 

4  pecks      -  1  bushel  bu. 

WINE  MEASURE. 

By  Wine  Measure  are  measured  Rum,  Brandyy  Per- 
ry, Cider,  Mead,  Vinegar  and  Oil. 

Its   denominations  are  pint,  quart,  gallon,  hogshead, 
pipe,  &c. 

2  pints  (ptf.)         make         1  quart         -         qt. 

4  quarts     -  1  gallon       -        gal, 

42  gallons  1  tierce        -         tier. 

63  gallons  1  hogshead  -         kfid. 

2  hogsheads        -  1  pipe  or  butt      P.  or  J3. 

2  pipes      -  1  ton  -  T. 

ALE,  OR  BEER  MEASURE. 

The  denominations  of  this  measure  are  pint,  quart, 
gallon,  barrel,  &c. 

4* 


[8  TABLE    OF    TIME    AND    MOTION. 

2  pints  (pt.)         make  1  quart     -  qt. 

4  quarts     -         -  1  gallon  -         -         gal. 

8  gallons   -         -         -  1  firkin  of  ale  -        fir. 

2  firkins     -  1  kilderkin       -         kil. 

2  kilderkins        -  1  barrel  -  bar. 

11  barrels,  or  54  gallons  1  hogshead  of  beer  hJid. 

2  barrels    -  1  puncheon      -        pun. 

3  barrels,  or  2  hogsheads     1  butt      -  butt. 

TIME. 

The  denominations  of  Time  are  year,  month,  week, 
tay,  hour,  minute  and  second. 
60  seconds  (sec.)        make         1  minute    -      min. 
60  minutes        -  1  hour        •      H. 

24  hours  -  1  day       '  -       D. 

7  days  -  1  week       -       W. 

52  weeks,  1  dav,  and  6  hours,  I   1  v 

or  365  days,  and  6  hours,  \   L  >'es 

12  months  (mo.)  1  year        -       Y* 
Note. — The  si#  hours  in  each  year  are  not  reckoned 

ill  they  amount  to  one  day;  hence,  a  common  year  con- 
sists of  365  days,  and  every  fourth  jrear,  called  leap 
year,  of  366  days. 

The  following  is  a  statement  of  the  number  of  days 
in  each  of  the  twelve  months,  as  they  stand  in  tho  cal- 
ender or  almanack : 

The  fourth,  eleventh,  ninth,  and  sixth, 
Have  thirty  days  to  each  affix'd : 
And  every  other  thirty-one, 
Except  the  second  month  alone, 
Which  has  but  twenty-eight  in  fine, 
Till  leap  year  gives  it  twenty-nine. 


MOTION. 

60  seconds  w    make          ]  prime    minute, 
60  minutes        -  1  degree      -  ° 

30  degrees  -         1  si^n  9. 

12  signs,  or  360  degrees  >          n^ 


REDUCTION. 


Reduction  teaches  to  change  numbers  of  one  denomi- 
nation into  those  of  other  denominations,  retaining  the 
same  value.  Its  operations  are  performed  by  Multipli- 
cation and  Division.  When  performed  by  Multiplica- 
tion, it  is  called  Reduction  Descending,  when  performed 
by  Division,  it  is  called  Reduction  Ascending. 

QUESTIONS. 

1.  How  many  farthings  will  it  take  to  make  two 
pence  ?    How  many  pence  to  make  two  shillings  ? — How 
many  shillings  to  make  two  pounds? 

2.  How  many  gills  to  make  three  pints? — How  many 
pints  to  make  three  quarts? — How  many  quarts  to  make 
three  gallons? 

3.  How  many  quarts  to  make  four  pecks? — How  ma- 
ny pecks  to  make  four  bushels? 

4.  How  many  pence  are  there  in  eight   farthings? — 
How  many  shillings  in  twenty-four  pence? — How  many 
pounds  in  forty  shillings? 

5.  How  many  pints  in  twelve  gills? — How   many 
quarts  in   six  pints? — How  many   gallons   in  twelve 
quarts? 

6.  How  many  pecks  in  thirty-two  quarts? — How  ma- 
ny bushels  in  sixteen  pecks? 

7.  How  many  pounds  and  shillings  in  thirty  shillings? 
How  many  shillings  and  pence  in  thirty  pence? 


REDUCTION  DESCENDING. 

RULE. 

Multiply  the  numbers  in  the  highest  denomination 
given,  by  the  number  that  it  takes  of  the  next  less  de- 
nomination to  make  one  of  that  greater;  and  thus  pro- 
ceed until  you  shall  have  multiplied  each  higher  de- 
nomination by  the  number  that  it  takes  to  form  the 
next  lower,  until  you  come  to  the  lowest  of  all. 


REDUCTION. 


PROOF. 


Descending  and   Ascending  Reduction  prove  each 
other. 


SIMPLE    EXAMPLES. 
I. 

Reduce  25  pounds  to  shillings 
25 
20  shillings  in  a  pound. 


500  shillings.  Ans.  500  shillings. 


n. 

Reduce  50  shillings  to  pence. 
50 
12  pence  in  a  shilling, 


600  pence.  Ans.  6CO  pence, 


in. 

Reduce  15  pence  to  farthings. 
15 
4  farthings  in  a  penny. 

60  farthings.  Ans.  60  farthings 


IV. 

Reduce  10  tons  to  hundred  weights. 
10 
20  hundreds  in  a  ton. 


200  hundreds.  Ans.  200  cwt 


REDUCTION.  51 


V. 

Reduce  36  pounds  to  ounces. 
36 
16  ounces  in  a  pound. 


216 

3G 

576  ounces.  Ans.  576  ounces. 

6.  Reduce  70  miles  to  furlongs  Ans.  560  fur. 

7.  Bring  30  furlongs  to  rods.  Ans.   1200  rods. 

8.  Bring  20  rods  to  feet.  Ans.  330  feet. 

9.  Bring  24  feet  to  inches.  Ans.  288  inches 

10.  Reduce  32  acres  to  roods.  Ans.   128  roods. 

11.  Bring  24  square  perches  to  square  yards. 

Ans.  726  square  yards. 

12.  Reduce  10  hogsheads  to  gallons.     Ans.  630  gal. 

13.  Bring  25  gallons  to  pints.  Ans.  200  pints. 

14.  Reduce  23  bushels  to  pecks.          Ans.  92  pecks, 

15.  Bring  12  pecks  to  pints.  Ans.  192  pints 

16.  Reduce  15  years  to  months.      Ans.  180  months 

17.  Bring  75  days  to  hours.  Ans.  1800  hours 

18.  Bring  24  hours  to  minutes.     Ans.   1440  minutes 

19.  Bring  10  signs  to  degrees.        Ans.  600  degrees 

COMPOUND    EXAMPLES. 
I. 

£.  s.  d.  qrs. 

In  15          17          11  3  how  many  farthings ! 

20  shillings  in  a  pound. 

317  shillings. 
12  pence  in  a  shilling. 


3815  pence. 

4  farthings  in  a  penny. 


15263  farthings. 


52  REDUCTION. 

Note. — In  multiplying  by  20,  I  added  in  the  17  shil- 
lings, by  12,  the  1 1  pence ;  and  by  4,  the  3  farthings ; 
and  this  must  be  observed  in  all  similar  cases. 

To  prove  this  sum,  let  the  order  of  it  be  changed, 
and  it  will  stand  thus:  in  15263  farthings  how  many 
pounds  ? 

4)15263 

12)3815+3  quarters. 


2|0)31  |7-}-ll  pence. 

£15  17s.  11  d.  3  qrs.  Ans. 

In  reducing  Federal  Money  from  a  higher  to  a  lower 
denomination,  it  is  only  necessary  to  annex  as  many  cy- 
phers as  there  are  places  from  the  denomination  given 
to  that  required ;  or  if  the  given  sum  be  of  different 
denominations,  annex  the  figures  of  the  several  denom- 
inations in  their  order,  and  continue  with  cyphers,  when 
the  sum  requires  it,  to  the  denomination  intended. 

2.  Thus,  in  7  eagles,  3  dollars,  how  many  mills? 

Ans.  73000. 

3.  In  85  dollars,  how  many  mills?  Ans.  85000. 

4.  In  574  eagles,  how  many  dollars?          Ans.  5740. 

5.  In  469  dollars, how  many  cents?         Ans.  46900. 

6.  In  844  dollars,  75  cents,  how  many  mills? 

Ans.  844750. 

7.  In  1000  dollars,  how  many  mills?     Ans.  1000000. 

8.  In  25  dollars,  47  cents,  8  mills ;  how  many  mills  ? 

9.  In  29  guineas  at  28s.  each,  how  many  pence  ? 

Ans.  9744. 

10.  In  20  acres,  29  poles,  or  perches,  how  many  square 

perches?  Ans.  3229. 

11.  How  many  solid  feet  in 30  cords  of  wood? 

Ans.  3840. 

12.  How  many  grains  in  100  Ibs. — Troy  Weight? 

Ans.  576000. 

13.  How  many  Ibs.  in  a  ton: — Avoirdupois  Weight? 

Ans.  2240. 


REDUCTION.  53 

14.  In    27  Ibs. — Apothecaries    Weight;    now    many 
grains?  Ans.  155520. 

15.  In  30  yards,  how  many  nails?  Ans.  480. 

16.  In    360  degrees,   being  the  distance   round   the 
world,  how  many  inches,  allowing  69i   miles  to  a  de- 
gree? Ans.  1,587,267,200. 

17.  How  many  pints  are  there  in  one  tun  of  wine? 

Ans.  2016. 

18.  How  many  half  pints  in  one  hogshead  of  beer? 

Ans.   864. 
19      How  many  pints  in  400  bushels?      Ans.  256CO. 

20.  How  many  seconds  in  80  years  of  365  days  each  ? 

Ans.  2,522,880,600 

21.  How  many  yards  in  4567  miles?      Ans.  603 "A&O. 

22.  In  £20  17s.,  how  many  pence  and   half  pence? 

Ans.  5004  pence,  and  10,008  half  pence. 

REDUCTION  ^ASCENDING. 

RULE. 

Divide  the  figure  or  figures  in  the  lowest  denomina- 
tion, by  so  many  of  that  name  as  make  one  of  the  next 
higher;  and  continue  the  division  until  you  have  brought 
it  into  that  denomination  which  your  question  requires. 

In  reducing  Federal  Money  from  a  lower  to  a  higher 
denomination,  nothing  more  is  necessary  than  to  cut  off 
so  many  places  on  the  right  hand  side  of  the  sum,  as 
there  are  denominations  lower  than  the  one  required. 
Thus,  98765  mills  are  reduced  to  dollars,  cents,  and 
mills,  by  cutting  off  one  figure  for  mills,  two  more  for 
cents,  and  the  remaining  figures  being  dollars,  the 
amount  is  $98|76|5 — or  ninety-eight  dollars,  seventy-six 
cents,  five  mills. 

SIMPLE    EXAMPLES. 

1.  How  many  dollars  are  there  in  8000  mills? 

8|00|0  Ans.  8. 

2.  In  487525  cents,  how  many  dollars  and  cents? 

4875|25  Ans.  $4875.25. 


54  REDUCTION. 

3.  In  999888  mills,  how  many  dollars,  cents    and 
mills?  999|8S|8  Ans.  $999.88.8. 

4.  In  19200  farthings,  how  many  pounds? 

4)19200 

12)4800 
20)400 


Ans.  20    pounds. 

5.  In  480  nails,  how  many  yards? 

4)480 

4)120 

30  Ans. 

COMPOUND    EXAMPLES. 

6.  In  52300  farthings,  how  many  pounds? 

4)52300 

12)13075 
2|0)108|9+7 


Ans.  £54  9s.  7d. 

7.  In  8828  Ibs.  Avoirdupois  Weight,  how  many  tons? 

Ans.  3  tons.  18cwt.  3qrs".  81bs. 

8.  In  524  Ibs.  Avoirdupois  Weight,  how  many  cwt. 
&c.  Ans.  4  cwt.  2  qrs.  *201bs. 

9.  In  253440  grains,  Troy  Weight,  how  many  Ibs? 

Ans.  44. 

10.  In  155520  grains,  Apothecaries  WTeight,  how  many 
pounds?  Ans.  27. 


REDUCTION, 


11.  How  many  miles  are  therein  1,585,267,200  in- 
ches? Ans.  25020. 

12.  In  4000  nails,  how  many  yards?  Ans.  250. 

13.  In  8000  square  rods,  how  many  acres?      Ans.  50. 

14.  In  2016  pints  of  wine,  how  many  tuns?     Ans.  1. 

15.  How  many  bushels  are  there  in  80,000  quarts? 

Ans.  2500. 

16.  In  2,522.880,000  seconds,  how  many  days? 

Ans.  29,2CO. 

17.  In  3840  solid  feet,  how  many  cords?         Ans.  30. 

18.  In  1728  half  pints  of  beer,  how  many  hogsheads? 

Ans.  2. 

19.  Bring  240,000  pence  to  pounds.         Ans.  £1000. 

20.  Bring  112  quarters  to  cwt.  Ans.  28  cwt. 

21.  Bring  120  miles  into  leagues.  Ans.  40L. 

22.  Bring  1280  poles  into  furlongs.          Ans.  32  fur. 

23.  Reduce  960  nails  to  quarters.  Ans.  240  qrs. 

24.  Reduce  17280  cubick,  or  solid  inches,  to  solid  feet. 

Ans.  10  solid  feet. 

25.  In  768  pints,  how  many  bushels?  Ans.  12. 

26.  In  1890  gallons,  how  many  hogsheads?     Ans.  30. 

Q.  1.  What  does  Reduction  teach? 

2.  By  what  rules  are  its  operations  performed  ? 

3.  When   performed  by  multiplication,  what  is  it 

called? 

4.  What  is  your  rule  for  Reduction  Descending? 

5.  When  performed  by  Division  what  is  it  called? 

6.  What  is  your  rule  for  Reduction  Ascending? 

7.  How  do  you  reduce  Federal  Money  frorn  a,  lower 

to  a  higher  denomination? 

8.  How  is  Reduction  proved' 


56 
COMPOUND  ADDITION. 


Compound  Addition  teaches  to  add  numbers  which 
represent  articles  of  different  value,  as  pounds,  shillings, 
pence;  or  yards,  feet,  inches,  &c.  called  different  de- 
nominations. The  operations  are  to  be  regulated  by 
the  value  of  the  articles,  which  must  be  learned  from 
the  foregoing  table. 

RULE. 

Place  the  numbers  to  be  added  so  that  those  of  the 
same  denomination  may  stand  directly  under  each  other. 
Add  the  figures  of  the  first  column  or  denomination  to- 
gether, and  divide  the  amount  by  the  number  which  it 
takes  of  this  denomination  to  make  one  of  the  next  high 
er.  Set  down  the  remainder,  and  carry  the  quotient  to 
the  next  denomination.  Find  the  sum  of  the  next  col- 
umn or  denomination,  and  proceed  as  before  through 
the  whole,  until  you  come  to  the  last  column,  which  must 
be  added  by  carrying  one  for  every  ten  as  in  Simple 
Addition. 

EXAMPLES. 

I.  ii.  in. 

£    s.    d.    qrs.  £  s.    d.    qrs.  £.  s.    d.  qrs. 

14  10    8      2  19  19  11     3  18  17  11  3 

11   16  10      1  10  14    4      1  15  14  10  3 

8     3  11      3  13  13  10     2  17  18     9  2 


34  11    6      2        44     8     2     2         52  11     8     OAns 


In  the  first  of  the  above  examples,  I  begin  with  the 
right  hand  column,  or  that  of  farthings;  and  having  ad 
ded  it,  find  that  it  contains  6.  Now,  as  6  farthings  con 
tain  1  penny  and  2  over,  I  set  the  2  farthings,  under  the 
column  of  farthings,  and  carry  the  penny  to  the  column 
of  pence.  In  the  column  of  pence  I  find  29,  which,  with 


COMPOUND     ADDITION.  57 

the  one  carried  from  the  farthings,  make  30.  In  30 
pence  I  find  there  are  2  shillings  and  6  pence  over :  set- 
ting the  6  pence  under  the  column  of  pence,  I  add  the  2 
shillings  to  the  column  of  shillings.  In  this  column  are 
29,  and  the  2  added  make  31.  Thirty-one  shillings  con- 
tain 1  pound, and  11  shillings  over.  The  11  shillings  are 
then  placed  under  the  column  of  shillings,  a.rA  the  1  is 
carried  to  the  column  of  pounds.  In  that  column  are  33 
pounds,  which,  with  the  1  added,  make  34.  Thus  the 
amount  of  the  sum  is  34  pounds,  11  shillings,  6  pence, 
and  2  farthings. 

In  ail  cases  in  Compound  Addition,  one  must  be  car- 
ried for  the  number  of  times  that  the  higher  denomina- 
tion is  contained  in  the  column  of  the  lower  denomination. 
Thus,  in  Troy  Weight :  as  24  grains  make  one  penny- 
weight, one  from  the  column  of  grains  is  carried  for  every 
24;  in  the  column  of  pennyweights,  one  for  every  20; 
and  in  every  instance  the  learner  must  be  guided  by  the 
foregoing  table  of  "Money,  Weights,  Measures,  &c. 

ir.  v. 

£      s.      d.      qrs.  £.      s.      d.      qrs. 

487    16     11        3  9876    15      4         3 

830    10       9        1  2123    14      5        0 

500    11       4        2  6789    18    10         2 

620    18       3        3  1234    15    II        '-1 

900      8     10        0  7876      493 


Note. — Sums  in  Compound  Addition  may  be  proved 
in  ihe  same  manner  as  in  Simple  Addition. 

TROY  WEIGHT. 

VI.  VII. 

Ib.  oz.  dwt.  gr.  Ib.  oz.  dwt.  gr. 

487  10  18  22  6780  11  11  12 

500  8  11  10  1100  9  18  22 

234  11  10  16  3090  10  17  20 

876  3  17  23  2468  8  13  19 


58                                 COMPOUND  ADDITION. 

AVOIRDUPOIS  WEIGHT. 

VIII.  IX. 

Ton.  cwt.  qr.  Ib.  oz.  dr.  Ton.  cut.  qr.  Ib  oz 

16  18  2  25  11  14  27  17  3  27  8 

97  12  3  17   9  11  98  19  2  11  <j 

34  11  1  10  10  10  70  11  1  18  7 

82  19  2  27  15  13  18  16  0  10  6 


APOTHECARIES  WEIGHT. 

x.  xi. 

fc     3     3  €  gr.  fc     3     3  €  gr. 

74     9    7  1  13  20  10     7  1  18 

18  11     6  2  17  37  11     5  2  17 

91   10    3  0  10  28     9     3  1  15 

17     9    5  1  19  14    8    4  0  11 


LONG  MEASURE. 

XII.  XII. 

deg.  mil.  fur.  po.  ft.  in.  mil.  fur.  po.  yd.  ft. 

118  36  7  19  13  3  976  2  13  4  2 

921  15  4  16  10  10  867  6  10  3  1 

671  10  6  27  11  11  500  1  11  0  0 

643  26  5  15  8  8  123  4  15  3  2 

123  14  5  16  7  8  345  6  17  1  0 


CUBICK,  OR  SOLID  MEASURE. 

XIV.  XV.  XVI. 

Ton  ft.  in.  yd.  ft.  in.  Cord  ft.  in. 

17  10  1229  29  20  1092  48  120  1630 
24  13  1460  11  11  1195  54  110  1500 
98  25  1527  18  11  1000  75  88  1264 

18  16  1079  27  9  1330  87  113  1128 


COMPOUND  ADDITION.             59 

LAND 

OR  SQUARE  MEASURE. 

XVII. 

XVIII. 

XIX. 

acr.  TOO.  per. 
987  2  23 

acr.  roo.  per. 
8423  1   38 

acr.  roo.  per. 
9432  3  24 

798  3  .28 

1234  0   10 

4324  2  12 

123  2  11 

4821  3   11 

5678   1  36 

567  1  27 

6789  2   30 

5865  3  11 

700  0  00 

8000  1   13 

8765  2  15 

CLOTH  MEASURE. 

XX. 

XXI. 

XXII. 

yd.  qr.  riL 
175  3  3 

El.  Fr.  qr.  nl. 
247   2  3 

El.  Fl.  qr.  nl. 
9876   2  3 

481  2  1 

456   1  1 

8765   1  2 

234  1  2 

345   3  0 

3456   2  3 

345  0  1 

236   2  2 

4000   0  0 

234  1  2 

567   0  1 

7898   2  3 

XXIII. 

XXIV. 

XXV.  , 

El.  E.  qr.  nl 

87654  1  2 

yd.  qr.  nl. 
656547  1  1 

yd.    qr.  nl. 
987654321  3   3 

56788  3  1 

987654  2  0 

234567876  0   0 

87654  3  2 

765432  1  3 

543212345  3   2 

12345  0  0 

134545  3  2 

900087654  1   3 

84231  2  3 

584050  0  1 

384563200  3   0 

DRY  MEASURE 

XXVI. 

XXVII. 

XXVIII. 

buxh.  pk.  qt. 
187  7  3 

bush.  pk.  qt. 
356  3  7 

bush.  pk.  qt. 
874   3   6 

290  6  2 

120  1  6 

123   1   2 

185  3  1 

543  2  1 

345  2   5 

349  1  2 

678  3  5 

753   1   7 

160  5  3 

432  1  3 

936  2   4 

5* 

60  COMPOUND    ADDITION. 

WINE  MEASURE. 

XXIX.  XXX. 

Tun.  khd.  gal,  qt.  pt.  Tun.  hhd.  gal.  qt.  pt 

4820  1   16  2  1  987654  1  12  1  1 

9765  3   18  3  1  321234  3  15  0  1 

8645  2   19  1  0  125780  2  18  3  1 

5432  1   22  3  1  876531  2  27  1  0 

6787  1   10  1  0  248765  1  49  2  1 


ALE  OR  BEER  .MEASURE. 

XXXI.  XXXII. 

khd.  gal.  qt.  pt.  hhd.    gal.  qt^  pt^ 

4820  48  3  1  17819174  18  3  1 

8765  34  1  1  21350000  27  1  1 

9877  53  2  1  12168400  35  0  0 

1234  12  1  0  21346870  15  S  1 

5678  50  0  1  43212345  50  1  1 


TIME. 

XXXIII.  XXXIV. 

w.     d.     h.     m.     s.  y.  mo.  w.     d.  h.  m.  s. 

3     6    23     58  24  75  11  3      6  22  50  57 

3     5    20    49  57  18  10  2      5  16  16  15 

1     4    21     30  30  84  11  14  15  10  10 

3    2    13    53  53  40  9  1      0  00  00  00 

1     0    10     10  10  SO  10  1      1  11  11  11 


MOTION. 

XXXV.  XXXVI.  XXXVII. 

18°  54*  44^  26°  19X     15n  10*.  24°  53'  50" 

20     25   30  19  26      20  90     19  31 

87     30    10  50  15      19  39    23  42 

00     11    11  33  10      11  8     17    44  45 

27     29    34  12  34     31  7     10    20    10 


COMPOUND    SUBTRACTION.  t)l 

Q.  1.  What  does  Compound  Addition  teach? 

2.  How  do  you  place  the  different  denominations  in 

Compound  Addition? 

3.  How  do  you  proceed  after  placing  the  denomina- 

tions under  each  other? 

4.  What  do  you  observe,  in  carrying  from  one  de- 

nomination to  another,  that  is  different  from 
Simple  Addition? 

5.  How  is  Compound  Addition  proved? 


COMPOUND  SUBTRACTION. 

Compound  Subtraction  teaches  to  find  the  difference 
between  any  two  sums  of  different  denominations. 
RULE. 

Place  those  numbers  under  each  other  which  are  of 
the  same  denomination — the  less  always  being  below 
the  greater.*  Begin  with  the  least  denomination,  and 
if  it  be  larger  than  the  figure  over  it,  consider  the  up- 
per one  as  having  as  many  added  to  it  as  make  one  of 
the  next  greater  denomination.  Subtract  the  lower 
from  the  upper  figure,  thus  increased,  and  set  down  the 
remainder,  remembering,  that  whenever  you  thus  make 
the  upper  figure  larger,  you  must  add  one  to  the  next 
superior  denomination. 

PROOF. 

As  in  Simple  Subtraction. 

EXAMPLES. 

ENGLISH  MONEY, 
i.  ii. 

£  s.  d.  qrs.  £  s.  d.  qrs  £  s.  d.  qr. 
460  14  10  3  744  10  8  1  689  792 
320  10  8  2  398  18  10  3  37218  4  3 


140      4     2    1      345    11      9    S        316    943 


*By  this  is  meant  that  the  lower  line  of. figure  must  always  be  a  less 
sura  than  the  upper  1'me,  though  sorae  of  its  au^ller  denominations 
may  be  larger  man  those  immediately  above  theru,  in  the  upper  line 


'JOMPOUND    SUBTRACTION. 

The  first  example,  is  in  itself,  sufficiently  plain.  In 
the  second,  finding  the  upper  figure  smaller  than  the 
lower  one,  as  it  is  in  farthings,  and  ns  four  farthings  make 
a  penny,  I  suppose  four  added  to  the  upper  figure,  which 
makes  it  5.  Then  I  say.  3  from  5,  and  2  remain.  Placing 
the  2  underneath,  I  add  1  to  the  next  lower  figure,  name- 
ly, the  10,  which  thus  becomes  1 1 ;  and  as  the  8  standing 
above  is  less,  I  suppose  12  added  to  it,  which  makes  it 
20.  Taking  11  f;om  20,  0  remain.  Setting  the  9  un- 
derneath, and  adding  one  to  the  18,  it  becomes  19;  and 
as  the  upper  figure  is  smaller,  I  suppose  20  added  to  it, 
which  makes  it  30.  I  take  19  from  30,  and  11  remain. 
Placing 4he  1 1  underneath,  I  carry  one  to  the  next  figure, 
namely,  8;  and  then  proceed  as  in  Simple  Subtraction. 

TROY  WEIGHT. 

iv.  v. 

$.       oz.    dwt.  gr.            Ib.  oz.  dirt.  gr. 

947        5      13  16  876543  7  16     11 

123       10      18  20  549876  9  17     19 


AVOIRDUPOIS  WEIGHT. 

TI.  VII. 

Ton.  c\vt.  qr.  Ib.  Ton.  cu-t.  qr.  Ib.  oz.  dr. 
5  13  1  12  8  16  0  24  11  11 
1  11  3  16  6  18  2  26  12  13 


APOTHECARIES  WEIGHT. 

VIII.  IX. 

fc     5     3     €    gr.  ft     3     3     6    gr. 

44     7    5     1     12  87    4      1     0      10 

39     9    6     2     16  48  10     4     1      18 


COMPOUND  SU3TRACTION.  C 

LONG  MEASURE. 

X.  XI. 

deg.  mil,  fur.  po.  ft.  in.  deg.  mil.  fur.  po. 
85  53  7  16  10  10  95  10  3  12 
60  57  0  27  14  11  79  44  6  13 


CUBICK,  OR  SOLID  MEASURE, 
xn.         xin.         xiv. 
Ton  ft.   in.    yd.  ft.   in.    Cord  ft.  in. 
18  17  1040   40  10   940  '  874  110  1122 
11  21  1485   32  16  1080    499  120  1699 


LAND,  OR  SQUARE  MEASURE. 

XV.  XVI.  XVII. 

acr.  roo.  per.  acr.  roo.  per.  acr.  roo.  per. 
987  2  23  8423  1  36  9432  3  12 
798  3  28  4123  0  10  7324  2  24 


CLOTH  MEASURE. 

xviir.                  xxix.                xx.  xxi. 

yd.  qr.  nl.     E.  E.  qr.  nl.     E.  Fl.  qr.  nl.  E.  Fr.  qr.  nl. 

45    12        537     2    1         567     1     2  945     3    3 

29   3     1        409    3    3        389    21  730    5    2 


DRY  MEASURE. 

XXII.  XXIII.  XXIV. 

bush.  pk.  qt.         bush.  pk.  qt.  bush.  pk.   qt. 
74     1     1             230    0    a  56      1     1 

42    3    2  199    2     1  28     3    3 


64  COMPOUND   SUBTRACTION, 

WINE  MEASURE, 
xxv.  xxvr. 

Tun.  hhd.  gal.  qt.  pt.  Tun.  khd.  gal.  qt.  pt. 

482     1       16    1     1  654    2      12      1     0 

297    3      22    3     1  276    3      40     2     1 


ALE  OR  BEER  MEASURE. 

XXVII.  XXVIII. 

hhd.  gal.  qt.  pt.      hhd.   gal.  qt.  pt. 
8240  12  1   1    11917400  10  0   0 
1987  52  2   2    11654000  27  2   2 


TIME. 

XXIX.  XXX. 

w.     d.     h.     m.     s.               y.     mo.  w.  d.     h.    m.     s. 

8      2     12     42  30             20      10  1    4     10    27  37 

1      1     16    54  40              11     11  35    17    40   54 


MOTION. 

xxxi.  xxxn.  xxxm. 

16°  15'  35"       8s.  10°  10'  10"  7s.  8°  37'  47" 

12     45  48         6     15     50   30  4    11    44     55 


Application  of  the  two  preceding  rules. 

1.  A  B  &  C  purchased  goods  in  partnership.     A  paid 
12  pound?,  10  shillings  and  8  pence;  B  paid  124  pounds, 
16  shillings;  and  C  paid  8^  pounds  and  11   pence. — 
What  was  the  whole  amount  paid?     Ans.  £224  7s.  7d. 

2.  A  merchant  has  money  due  him : — from  one  man, 
587  pounds;  from  another,  420  pounds,  17  shillings  and 
6  pence;  from  a  third,  200  pounds;  and  from  a  fourth, 
978  pounds,  16  shillings  and  8  pence.     How  much  had 
he  due  in  all?  Ans.  £2186  14s.  2d. 


COMPOUND    MULTIPLICATION.  65 

3.  From  20  pounds,  take  12  pounds,  19  shillings  and 
3  farthings.  Ans.  £7  Os.  lid.  Iqr. 

4.  From  22  pounds,  take  19  shillings  and  1  farthing. 

Ans.  £41  Os.  lid.  3  qrs. 

5.  From  17  pounds,  take  9  pounds,  9  shillings  and  9 
pence.  Ans.  £7  10s.  3d. 

6.  A  has  paid  B  £7  2s.  3d.  £19  lls.  4d.  and  £17 
18s.  Sid.  on  account  of  a  debt  of  £30.     How   much 
remains  unpaid?  Ans.  £15  7s.  7id. 

7.  A  ropemaker  received  3  tons,  4  cwt.,  2  quarters, 
and  5  pounds  of  hemp;  of  which  he  made  into  cordage 
2  tons,  9  cwt.  and  1  quarter.     How  much  had  he  left? 

Ans.    15cwt.  Iqr.  51bs. 
Q.  1.  What  does  Compound  Subtraction  teach? 

2.  H  nv  do  you  set  down  Compound  Subtraction  ? 

3.  What  do  you  do  when  the  lower  denomination 

is  larger  than  the  one  that  is  above  it  ? 

4.  How  is  Compound  Subtraction  proved? 


COMPOUND  MULTIPLICATION. 

Compound  Multiplication  teaches  how  to  find  the  value 
of  any  given  number  of  different  denominations,  re- 
peated a  certain  number  of  times.  It  is  of  great  use  in 
finding  t^e  value  of  goods,  which  is  generally  done  by 
multiplying  the  price  by  the  quantity, 

CASE  I. 

When  the  quantity  or  multiplier  does  not  exceed  12. 

Set  down  the  price  of  1,  and  place  the  multiplier  un- 
der the  lowest  denomination ;  and  in  multiplying  by  itt 
observe  the  same  rules  for  carrying  from  one  denomina- 
tion to  another  as  in  Compound  Addition. 

PROOF. 

Double  the  multiplicand,  and  multiply  by  half  the 
multiplier :  or,  divide  the  product  by  the  multiplier. 


66  COMPOUND    MULTIPLICATION* 

EXAMPLES. 

ENGLISH  MONEY. 

1.  What  will  7  yards  of  cloth  cost,  at  £1  12s. 
per  yard  ? 


£11   10s.  3|d. 

In  this  example,  1  say  7  time  3  make  21 — that  is,  21 
farthings,  equal  to  five  pence  and  one  farthing.  I  set 
down  the  one  farthing  under  the  place  of  farthings,  and 
carry  the  five  pence  to  the  place  of  pence  saying,  7  times 

10  are 70,  and  5  make  75  pence — equal  to 6  shillings  and 
3  pence.     I  set  down  the  3  pence  under  the  pence  in  the 
sum  and  carry  the  6  shillings,  saying  7  times  12  are  84, 
and  6  make  SO  shillings,  equal  to  4  pounds  and  10  shil- 
lings.    Setting  down  the  10  shillings  under  the  shillings, 
I  carry  the  4  pounds,  saying  7  times  1  are  7,  and  4  make 

11  pounds,  making  thefanswer  to  the  question  1 1  pounds, 
10  shillings,  3  pence  and  1  farthing. 

II.  III.  IV. 

£    s.    d.         £    s.    d.         £    *.     d. 
Multiply    4    14    101      7     12    9         14    15     94 
by  2  4  8 

9    9    91 


V.  VI.  VII. 

£    *.  d.  £    *.    d.  £    *.    d. 

Multiply  14  17  84  24  16  10*  50     15  5f 

by  9  7  12 


TROY  WEIGHT. 

VIII.  IX. 

Z&.     oz.  diet.    gr.         Ib.     oz.  dwt.    gr. 
Multiply  11      9     16      10         17      5     12      6 
by  4  5 


COMPOUND  MULTIPLICATION.  67 

AVOIRDUPOIS  WEIGHT, 
x.  xi. 

Ton.  cwt.  qr.  Ib.  oz.  dr.       Ton.  cwt.  qr.  Ib.  oz.  dr. 
Mult.  3     11     3    10  5    4          6     17    3     13    2    15 
by  6  8 


APOTHECARIES  WEIGHT. 

XII.  XII. 

fc     3     3     6    gr.  ft     3     3     6    gr. 

Mult.  43    10    6    2    10  4     10    7     2    13 

by  5  6 


LONG  MEASURE. 

XIV.  XV. 

dcg.  m.  fur.  p.  yd.  ft.  in.  L.  m.  fur.  p. 

Mult.  7    22    6    20    2    2    10  15   2    7    30 

by  26 


CUBICK.  OR  SOLID  MEASURE. 

XVI.  XVII.  XVIII. 

Ton.  ft.  in.         yd.  ft.  in.          Cord  ft.  in. 
Mult.  10   16  15         14   2    19          24     13    18 
by  2  4  0 


LAND,  OR  SQUARE  MEASURE. 

XIX.  XX.  XXI.  ^ 

A.  R.  P.    A.  R.  P.    A.  R.  P. 

Mult.  20    3     12         37    2    15          47     1     18 
bv  2  4  6 


68  COMPOUND    MULTIPLICATION. 

CLOTH  MEASURE. 

XXII.  XX1TI.  XXIV.  XXV. 

yd.  qr.  nl.  ELFr.  qr.  nl.  El.  Fl.  qr.  nl.  El.  E.  qr.  nl 

MulU7  33        32    2     1         42    2    1       53     2     1 

by  4  6  8  9 


bush.  pk.  qt. 
Mult.     637 
by  5 


DRY  MEASURE. 

XXVII. 

bush.  pk.  qt. 
14     3     2 
6 


XXVIII. 

bush.  pk.  qt. 
34      2     3 

8 


WINE  MEASURE. 

XXIX.  XXX. 

Tun.  hhd.  gal.  qt.  pt.  Tun.  hhd.  gal.  qt.  pt, 

Mult.    1       2       12 


by 


3     1 

4 


40     3     1 
10 


ALE,  OR  BEER  MEASURE. 

XXXI.  XXXII. 

hhd.  gal.  qt.  pt.  hhd.  gal.  qt.  pt. 

Mult.  3     12     2     1  4     15     3     1 

by  5  8 


XXXIII. 

y.  mo.  w.  d, 
Mult.  7     735 
by  9 


XXXV. 

Mult.  24°     19'     11" 
by  10 


TIME. 

XXXIV. 

y.  mo.  w.    d.     h.     m.     s. 
8    5    3    6    20    32    10 

7 


MOTION. 

XXXVI. 

10s.     30°     17'     101" 
12 


C03IPOUKD    MULTIPLICATION.  69 

CASE  II. 

When  the  multiplier  or  quantity  exceeds  12,  and  is  the  pro- 
duct of  two  factors  in  the  Multiplication  Table;  that  w, 
of  two  numbers  which  being  multiplied  together,  amount 
to  the  same  as  the  multiplier. 
Multiply  the   sum  by   one  of  the  two  numbers,  and 

then  multiply  the  product  by  the  other. 

EXAMPLES. 

I.  II. 

£      8,      d.  £      S.      d. 

Multiply  8    18    111  by  18.  1312    9*  by  27. 

6  9 


53  13  10*  122  15  1* 

3  3 


161     1  74  368    5  44 


£    s.  d.  £  s.    d. 

3.  Multiply  10  10  10  by  14.    Product  147  11     8 

4.  "       11   11   11  by  15.  «  173  18     9 

5.  «       12  12     9  by  24.  «  303  6    0 

6.  "         5  13     44  by  28.  «  158  14    6 

7.  "         4  15  10  by  42.  "  201  5     0 

8.  «         7  17     71  by  64.  «  504  9     4 

9.  «        6  10     3  by  72.  "  468  18  0 

10.  "        9  19  Hi  by  81.  «  809  16  74 

11.  «       10  15    91  by  84.  «  906  8  3 

12.  «         3  11     74  by  96.  «  343  16  0 

CASE   III. 

When  the  quantity,  or  multiplier,  is  such  a  number  that  no 
two  numbers  in  the  Multiplication  Table  will  produce  it. 
Multiply  the  sum  by  two  numbers  whose  product  will 
amount  to  nearly  the  same  as  the  multiplier;  then  mul- 
tiply the  sum  by  the  number  which  will  make  the  pro- 
duct of  the  two  numbers  equal  to  the  multiplier,  and 
add  its  product  to  the  sum  produced  by  the  two  num- 
bers. 


70  COMPOUND    MULTIPLICATION. 

EXAMPLES. 
I. 

£.    ,.    d.  £.    ..    d. 

Multiply  7     10    5  7     10    5 

by  62  10  2 

75    4    2                           15     0  10 
6  


451  5     0        He  re  note,  I  multiply  by  10,  then 
15  0  10     by  6,  because  10  times  6  make  60; 

then  I  multiply  the  same  sum  by 

468  5  10     2,   that  I  multiplied,  first,  by  10, 

and  add  its  product  to  the  other 

product,  which  makes  the  amount  of  the  answer. 

£    s.    d.  £   s.    d. 

2.  Multiply  2    10  10     by  31.  Product  18  15  10 

3.  "  3    11   11  by  38.  "  136  12  10 

4.  «  4    11     24  by  68.  «  310    0    9 

5.  "  1      8     8  by  26.  «  37     5    4 

6.  "  1      3     3*  by  47.  "      54  14    Si 

7.  «  124*  by  83.  «       92  17     li 

CASE  IV. 

When  the  multiplier  is  greater  than  the  product  of  any  two 

numbers  in  the  Multiplication  Table. 
Multiply  the  sum  by  10,  and  thatproductby  10,  which 
is  equal  to  multiplying  by  100;  then  multiply  the  pro 
du-^t  by  the  number  of  hundreds  in  the  multiplier,  and  il 
the  sum  be  even  hundreds,  the  product  will  t  e  the  answer. 
If  (here  be  odd  numbers  over  even  hundreds, as 70, 60,  or 
37,  &c.,  multiply  the  amount  or  product  of  the  first  mul 
^plication  by  10,  by  the  number  of  tens  over  100;  thus, 
if  there  be  70  over,  multiply  by  7.  If,  in  additon  to 
tens,  there  are  smaller  numbers,  as  7,  8, 5,  &c.,  the  sum 
must  be  multiplied  by  such  number;  and  the  amount  of 
all  the  multiplications  being  then  added  together,  their 
sum  will  be  the  answer. 


COMPOUND   3HTJLTIILICATIOX.  71 

EXAMPPLES. 
1. 

£        s.        d. 

Multiply  1         2        4 

by  4300  10 

11         3        4  amount  of  10. 
10 


111       13         4  amount  of  100. 
10 


1116       13        4  amount  of  1000. 
4 


4468       13        4  amount  of  4000. 
335       00         0  amount  of  300. 


4801       13        4  Answer. 


In  the  foregoing  example,  I  first  multiply  by  10,  tbree 
imes,  which  gives  the  amount  of  the  sum  multiplied  by 
lOOOf  then  by  4,  which  gives  the  amount  of  00  \ — 
The  s\imis<yet  to  be  multiplied  by  300.  To  do  this,  1 
take  the  product  of  thssum  multiplied  by  100,  viz.  111Z. 
13s.  4d.  and  multiply  it  by  3,  which  gives  the  product 
of  the  sum  by  300.  The  sum  of  these  is  the  answer. 
£  s.  d.  £  s.  d. 

2.  Multiply         1      4  by  190.     Product    12  13     4 

3.  «         1     2      3  by  430.  "       478     7     6 

4.  "  76  by  506.  «       189  15     0 

5.  «  8      8  by  684.  «      296    8     0 

6.  «         139  by  375.  «      445     6    3 

7.  «  1      2  by  3458.  «      201  12     0 

APPLICATION. 

1 .  What  do  84  pounds  of  sugar  cost  at  9d.  per  pound  ? 

Ans.  £3.  3s. 

2.  What  do  18  yards  of  cloth  cost  at  19s  per  y^rd. 

Ans.  £17.  2s. 

3.  Sold  7  tons  of  iron  at  £32    10s.    per   ton;   hi/v 
much  is  the  amount?  Ans.  £227.  10,?. 


COMPOUND    MULTIPLICATION. 

4.  What  is  the  weight  of  4  hogsheads  of  sugar,  each 
weighing  7  cwt.  3  qrs.  19  Ib?    Ans.  31  cwt.  2  qrs.  20  rbs. 

5.  What  is  the  weight  of  6  chests  of  tea,  each  weigh- 
ing 3  cwt.  2  qrs.  9  Ibs?  Ans.  21  cwt.  1  qr.  26  Ibs. 

6.  What  is  the  value  of  79  bushels  of  wheat,  at  11s. 
5}d.  per  bushel?  Ans.  £45  6s.  1Q4. 

7.  What  is  the  value  of  94  barrels  of  cider,  at  12s. 
2d.  per  barrel?  Ans.  £57  3s.  8d. 

8.  What  is  the  value  of  114  yards  of  cloth,  at  15s. 
d.  per  yard?  Ans.  £87  5s.  7£d. 

9.  What.is  the  value  of  12  cwt.  of  sugar,  at  £3  7s. 
4d.  per  c>vl.?  Ans.  £40  8s. 

10.  What  is  the  worth  of  63  gallons  of  oil  at  2s.  3d. 
per  gallon?  Ans.  £7  Is.  9d. 

11.  What  is  the  amount  of  120  days  wages  at  5s.  9d. 
per  day?  Ans.  £34  10s. 

12.  What  is  the  worth  of  144  reams  of  paper  at  13s. 
4d.  per  ream?  Ans.  £96. 

13.  What  will  1  cwt.  of  sugar  cost,  at  ICfd.  per  Ib?* 

Ans.  £5  Os.  4d. 

14.  If  I  have  9  fields,  each  containing  12  acres,  2  roods 
and  25  poles;  how  many  acres  have  I  in  the  whole? 

Ans.  113A.  3R.  25P. 

15.  What  will  1  ton  of  lead  cost,  at  1  pence  per  pound? 

Ans.  £37  6s.  8d. 

16.  If  a  man  can  travel  25  ms.  3  fur.  20  ps.  4  yds.  in  1 
day,  how  far  can  he  travel  in  6  days? 

Ans.  152ms.  5fur.  4ps.  2yds 
Q.  1.  What  t!c?*  Compound  Multiplication  teach? 

2.  In  what  is  it  par'i'-nlarly  useful? 

3.  Which  is  made  the  multipli^v— the  price,  or  the 

quantity  ? 

4.  Howr  do  you  proceed  when  the  multiplier  does  not 

exceed  12? 

5.  How  do  you  proceed  when  the  mul  tiplier  exceeds  12  ? 

6.  When  the  multiplier  consists  of  no  two  component 

numbers,  as  in  case  third,  how  do  you  proceed? 

7.  How  do  you  proceed  in  case  fourth? 

8.  How  is  compound  multiplication  proved? 

*It  must  be  recollected  that  Icwt.  is  U21h>. 


COMPOUND  DIVISION. 

Compound  Division  teaches  the  manner  of  dividing 
numbers  of  different  denominations. 

CASE  I. 

When  the  divisor  does  not  exceed  12. 

Begin  at  the  highest  denomination,  and  after  dividing 
that,  if  any  thing  remain,  reduce  it  to  the  next  lower  de- 
nomination, adding  it  to  that  denomination  in  the  sum,  and 
proceed  in  this  manner  until  the  whole  is  divided.  If  the 
number  of  either  denomination  should  be  too  small  to  con- 
tain the  divisor,  reduce  it  to  the  next  lower  denomina- 
tion, and  add  it  thereto,  as  directed  in  case  of  a  remain- 
der. The  denominations  in  the  quotient  must  be  kept 
separate. 

PROOF. 

Multiply  the  quotient  by  the  divisor,  and  the  product, 
if  right,  will  be  equal  to  the  dividend. 

EXAMPLES. 

I.  II.  III. 

£     s.     d.         £     s.     d.  £     s.     d. 

Divide  2)6      8    8       4)3     3     10        5)7    2      3 


344  15  Hi  18     54+3 


iv.  v.  vi. 

£    s.     d.  £     s.     d.  £     s.      d. 

5)6     17    11          6)9     9      9        12)21    16    lid 


177  1    11      7i  1      16     4*+10 


In  doing  the  6th  sum,  which  is  divided  by  12,  I  find 
the  divisor  is  contained  once  in  21;  and  setting  down  1 
I  find  9  pounds  remaining;  which,  reduced  to  shillings, 
and  added  to  the  16  shillings  in  the  sum,  make  196 
shillings.  The  divisor  being  contained  16  times  in  196, 
with  4  remaining,  I  set  clown  16,  and  reducing  the  4 
shillings  to  pence,  and  adding  them  to  the  11  pence,  in 
the  sum,  the  amount  is  59  pence.  The  divisor  is  con- 


74  COMPOUND   DIVISION. 

tained  4  times  in  59,  leaving  1 1  pence  remaining.  I  set 
down  4,  and  the  remaining  1 1  pence  reduced  to  farthings 
and  added  to  the  half  penny  or  2  farthings  in  the  sum, 
make  46  farthings ;  and  as  the  divisor  is  contained  3 
times  in  46,  leaving  a  remainder  of  10, 1  set  down  I  and 
place  the  final  remainder  at  the  right  hand  of  the  sum. 
£  s.  d.  £  *.  d. 

7.  Divide  12    10  10    by    5.  Quotient  2    10    2 

8.  «       13    13     9    by   4.         «        3      8     5* 

9.  «         2    18  Hi  by    3.         «  19    7£+l 

10.  «         7      7     7   by    4.         «        1     16  101 

11.  «     177    19  111  by  12.        «      14    16    71+11 

CASE    II. 

When  the  divisor  exceeds  12,  and  is  the  product  of  two 

numbers  multiplied  together. 
Divide  by  one  of  the  numbers:  then  divide  the  quo- 
tient by  the  other. 

EXAMPLES. 
Divide  £5  10s.  6d.  by  48. 

£    «!    d. 
6)5    10   6 

8)  18  5 

2   3+5  Answer. 

Note. — If  there  be  any  remainder  in  the  first  opera- 
tion, and  not  in  the  second,  it  is  the  true  one.  When 
there  is  a  remainder  in  the  second  operation,  multiply  it 
by  the  first  divisor,  and  add  it  to  the  first  remainder,  if 
there  be  any,  and  it  forms  the  true  remainder. 

£    s.     d.  £    s.    d. 

2.  Divide  240  12  10    by  16.  Quotient  15    0    9i+8 

3.  «  88  13  11    by  21.  «  44    51+14 

4.  "  90  15  4i  by  32.  «  2  16    81+2 

5.  «  450  8  8  by  42.  «  10  14  51+26 
"  789  19  9  by  64.  "  12  6  10i+52 
"  840  4  3*  by  72.  «  11  13  4d+62 


COMPOUND    DIVISION.  75 

CASE  III. 

When  the  divisor  is  more  than  12,  and  cannot  be  produced 

by  multiplying  any  two  numbers  together. 
Divide  after  the  manner  of  Long  Division,  reducing 
from  higher  to  lower  denominations,  as  in  the  following 

EXAMPLES. 

Divide  £61  12s.  by  23.  Sfc 

i. 

£     *. 

23)61     12(£2  13s.  6d.  3qrs.+3  Ans. 
46 

Divide  £14  10s.  llfd.  by  95. 
15  ii. 

30X  £     *.     d. 

95)14     10     1  H(£0  3s.  Of  d.4-2  Ans. 
312  20  X 

23  - 

—  290 

82  285 

69  - 

—  5 

13  12X 

12X  - 

-  71 

156  4x 

138  - 


18  28 

4X 

—  2 

72 

69 


Note. — In  the  second  example,  I  find  the  divisor  great- 
er than  the  number  of  pounds  in  the  dividend.  I  there- 
fore set  down  a  cypher  in  the  place  of  pounds  in  the 
quotient,  then  reduce  the  14  pounds  in  the  sum,  into 
shillings,  at  the  same  time  adding  the  ten  shillings  in  the 


70  COMPOUND    DIVISION 

sum,  which  thereby  becomes  290.  In  200  the  divisor 
is  contained  3  times,  and  5  over.  This  5  shillings  I  re- 
duce to  pence,  adding  to  it  the  1 1  pence  in  the  sum ;  and 
the  amount  being  still  smaller  than  the  divisor,  I  set 
down  a  cypher  in  the  place  of  pence,  in  the  quotient,* 
and  reduce  it  to  farthings,  and  proceed  as  before. 

Though  this  operation  is   longer,  it  is,  perhaps,  less 
liable  to  error  than  either  of  the  preceding  cases. 
£     s.     d.  £  ;*.     d. 

3.  Divide  20    10    8    by  17.    Quotient  1     4     l£4-9 

4.  «  27    18    7  by  29.  «  0  19  3  +4 

5.  «  147     4    4  by  65.  «  25  3 » -4-18 

6.  "  581  19  11 1  by  73.  «  7  19  5J4-49 

7.  «  77     3    3J  by  19.  "  4     1  2'+17 

8.  «  319    7  lOj  by  29.  «,  11     0  3J+1 

APPLICATION. 

1 .     If  42  cows  cost  £126  16s.  6d ;  what  was  the  price 
of  each?  Ans.  £3  Os.  4'd. 

3.  Five  men  bought  a  quantity  of  hay,  weighing  21 
tons,  13  hundred,  and  3  quarters;  which  they  divided, 
equally  among  them.     What  was  the  share  of  each? 

Ans.  4  tons,  6cwt  3qrs. 

4.  A  farmer  had  3  sons,  to  which  he  gave  a  tract  of 
land  containing  520  acres, 3  roods,  29  perches;  and  the 
land  was  to  be  divided  equally  among  them.     What  was 
the  portion  of  each?  Ans.  173A.  2R.  23P. 

5.  Divide  375  miles,  2  furlongs,  7  poles,  2  yards,  1 
foot,  2  inches,  by  39.      Ans.  9M.  4fur.  39P.  Oyd.  2ft.  Sin. 
Q.     1 .     What  does  Compound  Division  teach  ? 

2.  How  do  you  proceed  when  the  divisor  does  not 

exceed  12? 

3.  How  ?  When  the  number  of  either  denomination 

is  too  small  to  contain  the  divisor  ? 

4.  How?  When  the  divisor  exceeds  12.  and  is  the 

product  of  two  numbers  multiplied  together? 

5.  How?  When  the  divisor  is  more  than  12,  and 

cannot  be  produced  by  multiplying  any  two 
numbers  together? 

6.  How  is  Compound  Division  proved? 


77 

EXCHANGE. 

Exchange  teaches  to  change  a  sum  of  one  kind  of 
money  to  a  given  denomination  of  another  kind. 

To  reduce  the  currency  of  each  of  the  United  States  tc 
dollars  and  cents,  or  Federal  Money. 

RULE. 

Reduce  the  sum  to  pence;  to  the  pence  annex  twc 
cyphers ;  then  divide  by  the  number  of  pence  in  a  dol 
lar,  as  it  passes  in  each  State,  the  quotient  or  answer 
will  be  in  cents,  which  may  be  easily  reduced  to  dollars 

Note. — This  rule  applies  to  the  currency  of  any  other 
country,  if  its  currency  be  in  pounds,  shillings,  pence 

c. 

EXAMPLES. 

1.  Reduce  621  pounds,  New  England  Virginia,  anc 
Kentucky  currency,  to  dollars  and  cents;  a  dollar  being 
72-  Dence. 

£ 
621 
20 


72)14904000($2070.00 
144 

504 
504 

000 

2.  Reduce  12  pounds,  3  shillings,  and  9  pence  to 
dollars  and  cents.  Ans.  $40.62*. 

3.  Reduce  30  pounds  and  3  shillings  to  dollars  and 
cents.  Ans.  $100.50. 

4.  In  £763  how  many  dollars,  cents  and  mills? 

Ans.  $2543.33cts.  3  m. 

5.  Reduce  19  shillings  and  10  pence  to  dollars  and 
•ents.  Ans.  $3.30cts.  5m. 


78  EXCHANGE. 

6.  In  9  pounds  and    16  shillings  in  New  York  and 
Vorth  Carolina  currenncy,  how  many  dollars  and  cents, 
•eckoning  96  pence  to  a  dollar?  Ans.  $24.50. 

7.  In  30  pounds,  how  many  dollars  and  cents,  same 
currency?  Ans.  75.00. 

8.  In  27  pounds,  2  shillings,  how  many  dollars  and 
cents,  same  currency  ?  Ans.  67.75. 

9.  In  942  pounds  of  New  Jersey,  Pennsylvania,  Del- 
aware, and  Maryland  currency ;  how  many  dollars  and 
cents,  a  dollar  being  90  pence?  Ans.  $2512.00. 

10.  In  12  pounds  how  many  dollars  and  cents  same 
currency?  Ans.  $32.00. 

11.  In  86  6s.  5d.  how  many  dollars,  cents  and  mills, 
same  currency?  Ans.  $230.18  cts.  8m 

12.  In  21  pounds,  South  Carolina  and  Georgia  cur- 
rency,  how  many  dollars  and  cents,  there  being  56 
pence  in  a  dollar?  Ans,  $90.00 

13.  In  56  pounds,  how  many  dollars,  &c.  same  cur- 
rency? Ans.  $240.00 

14.  108  pounds,  Canada  and  Nova  Scotia  currency, 
low  many  dollars,  &,c.  there  being  60  pence  in  a  dollar  \ 

Ans.  $432.00 

15.  In  460  pounds  and  16  shillings  sterling,  or  Eng- 
ish  money,  how  many   dollars,  &c.   there  being  54 

pence  in  a  dollar?  Ans.  $2048.00 

16.  Reduce  16  pounds,  6  shillings,  and  3  pence  Eng- 
lish money,  to  dollars  and  cents.  Ans.  $72.5'~ 

To  bring  Federal  Money  into  pounds  shillings,  fy  pence 
RULE. 

Multiply  the  dollars,  or  dollars  and  cents,  by  the 
number  of-  pence  in  a  dollar  of  the  currency  to  which 
you  wish  to  change  the  given  sum ;  the  answer  will  be 
in  pence,  which  can  then  be  reduced  to  shillings  and 
pounds.  When  there  are  cents  in  the  sum  to  be  re 
duced,  two  figures  must  be  cut  off  from  the  right  of  the 
product,  before  bringing  it  into  pounds. 

Note. — This  rule  applies  to  the  currency  of  any 
country  whose  currency  is  in  pounds,  shillings,  &c. 


EXCHANGE.  79 

EXAMPLES. 

1.  In  $16.50  how  many  pounds  and  shillings,  in  ster- 
ling, or  English  money;  a  dollar  being  four  shillings 
and  six  pence,  or  54  pence? 

$16.50 
9X6=54  9 


12)891.00 

2|0)7|4.+3 
£3  14s.  3d.  Answer. 

2.  In  33  dollars,  how  many  pounds,  &c.  ? 

Ans.  £7.  8s.— 6d. 

3.  In  1000000  dollars,  how  many  pounds  sterling? 

Ans.  £225000.* 

4.  Reduce  432  dollars  into  the  currency  of  Canada 
and  Nova  Scotia,  a  dollar  being  equal  to  five  shillings; 
or  60  pence.  Ans.  £108. 

5.  In  $490.50  how  many  pounds,  shillings,  &,c.  same 
currency?  Ans.  £122  12s.  6d. 

6.  Bring  $150.25,  into  the  currency  of  New  England, 
Virginia,  and  Kentucky,  a  dollar,  being  equal  to  72 
pence.  Ans.  £45.  Is.  6d. 

Questions. 

1.  What  does  Exchange  teach? 

2.  How  do  you  reduce  the  currency  of  any  one  of 

the  United  States  to  Federal  Money? 

3.  Does  this  rule  apply  to  the  currency  of  any  other 

country? 

4.  How  do  you  change  Federal  Money  into  pounds, 

shillings,  and  pence  of  any  state  or  country? 

5.  Among  the  various  kinds  of  money,  what  kind  is 

the  most  easily  reckoned  ? 

*Federal  Money  may,  also,  be  changed  into  English  Money,   by 
multiplying  the  dollars  by  9,  and  dividing  the  product  by  40. 


to 
PLATE 

TO  BE  USED  IN  STUDYING  VULGAR  FRACTIONS. 

i 

i 

Jn  using  the  Fractional  Plate,  the  student  must  count  the  white 
spaces,  and  not  the  black  lines.     The  first  row   of  squares,  or  white 
spaces,  at  the  top,  are  whole  numbers;  the  second  row  is  divided  into 
lalves;  the  third,  into  thirds,  and  so  on  from  the  top  to  the  bottom. 
Thus  it  may  be  shown,  at  one  glance,  that  7  halves  make  three  and 
a  half,  or  that  8  thirds  make  2  and  2  thirds,  &c. 

VULGAR  FRACTIONS. 

Fractions  are  broken  numbers,  expressing  any  as- 
signable part  of  an  unit,  or  whole  number.  They  are 
represented  by  two  numbers  placed  one  above  another, 
with  a  line  drawn  between  them;  thus  f ,  I,  &,c.  signi- 
fying two  fifths  and  five  eights. 

The  figure  above  the  line  is  called  the  numerator, 
and  that  below  the  line,  the  denominator.  The  denom- 
inator shows  into  how  many  equal  parts  the  whole 
quantity  is  divided,  and  represents  the  divisor  in  divis- 
ion.— The  numerator  shows  how  many  of  those  parts 
are  expressed  by  the  fraction ;  being  the  remainder  af- 
ter division.  Both  these  numbers  are  sometimes  called 
the  terms  of  the  fraction. 


Questions  to  prepare  the  learner  for  this  Rule. 

1.  If   a  pear  be  cut  in  two  equal  parts,  what  is  one 
of  those  parts  called?  Ans.  a  half. 

2.  If  you  cut  a  pear  into  three  equal  parts,  what  is 
one  of  those  parts  called?  Ans.  one  third. 

3.  How  many  thirds  of  a  thing  make  the  whole? 

4.  If  a  pear  be  cut  into  four  equal  parts,  what  is  one 
of  those  parts  called?     Ans.  one  fourth.     What  are  two 
of   the  parts   called?     Ans.   two   fourths.     What  are 
three  of  them  called  ?  Ans.  three  fourths. 

5.  How  many  fourths  of  a  thing  make  the  whole? 

6.  If  an  orange  be  cut  into  five  equal  parts,  what  is 
one  of  the  parts  called?     Ans.  one   fifth.     What  are 
two  of  the  parts  called?     Ans.  two  fifths.     What  are 

hree  of  them  called?     Ans.  three  fifths.     What  are 
four  of  them  called?  Ans.  four  fifths. 

7.  How  many  fifths  of  a  thing  make  the  whole? 

8.  If  you  cut  a  pear  into  six  equal  parts,  what  is  one 
of  the  parts  called?     What  are  two  of  them  called? — 
What  are  three  of  them  called  ?    What  are  four  of  them 
called? 

9.  How  many  thirds  are  there  in  one  ?     How  many 
fourths?     If  four  fourths  make  the  whole,  what  part 
are  two  fourths?     What  part  of  three   is  one?     What 
part  of  four  are   two?     What  part  of  six  are  two? — 


82  VULGAR  FRACTIONS. 

What  part  of  eight  are  two  ?  What  part  of  eight  are 
six?  What  part  of  9  are  6?  What  part  of  10  are  2? 
I  What  part  of  10  are  4?  What  part  of  10  are  7?  What 
part  of  12  are  6?  What  part  of  12  are  4?  What 
part  of  12  are  3?  What  part  of  12  are  2? 

10.  How  many  are  two  fourths  of  12?    How  many 
are  three  fourths  of  12?     Two  thirds  of  12,  are  how 
many?     How  many  are  5  times  8?     In  one  eighth  of 
40,  how  many?     In   three   eights  of  40,   how  many? 
Four  eights  of  40,  are  how   many?     Then  $  of  any 
p-;mber,  or  of  any  thing,  amount  to  how  many,  or  how 
much?     How  many  are  f  of  30?     How  many  are  £  of 
30?     How  many  in  i  of  60?     In  *  of  60,  how  many? 
In  }  of  60,  how  many     In  ^  of  60,  how  many  T  How 
many  are  -fa  of  60?     How  many  are  f  of  60?'    How 
many  are  £  of  60?     In  2  and  J,  how  many  fifths?    In 
5  and  f  how  many  fifths?    In  }  of  100,  how  many? 
In  £  of  100  cents,  or  1  dollar,,  how  much?     How  much 
are  I  and  i?     How  much  are  I  and  f  ?     How  much  are 
i  and  |?     How  much  are  f?     How  much  are  f  and 
How  much  are  J,  |,  and  A?     Iny,  how  many?     In 
how  many?  If  you  take  I  from  one  dollar,  how  much 
will  remain?     If  you  take  J  from  one  dollar,  how  much 
will  remain?    If  you  take  A  from  a  pound,  how  much 
will  remain?     If  you  take  -|  from  one,  how  much  will 
remain  ?     How  many  fourths  are  2  times  I  ?  How  many 
are  5  times  f?    How   many  are  3  times  f?     In 
how  many  ?     In  -1—,  how  many  ? 

11.  If  one  half,  three  fourths  and  a  quarter,  be  add- 
ed, how  much  will  be  their  amount? 

12.  If  you  take  two  eights  from  eleven  eights,  how 
much  will  remain? 

13.  What  is  a  proper  fraction? 

Ans.  When  the  numerator  is  less  than  the  denomi- 
nator, as  i,  or  |,  &c. 

14.  What  is  an  improper  fraction? 

Ans.  It  is  that  in  which  the  numerator  is  equal,,  or 
superior  to  the  denominator;  as,  ^,  or  |,  or  J,  &c. 

15.  What  is  a  simple  fraction? 

Ans.  It  is  a  fraction  expressed  in  a      simple  form; 
as,  4,  A,  f 


VULGAR  FRACTIONS  85 

16.  What  is  a  compound  fraction? 

Ans.  It  is  the  fraction  of  a  fraction,  or  several  frac- 
ions  connected  together  with  the  word  of  between 
hem;  as  £,  of  I  of  I;  or  I  of  T7T,  &c.  which  are  read 
hus,  one  half  of  two  thirds;  &,c. 

17.  What  is  a  mixed  number? 

Ans.  It  is  composed  of  a  whole  number  and  a  frac- 
ion;  as  3i,  or  12i. 

18.  What  is  the  common  measure  of  two  or  more 
lumbers? 

Ans,  It  is  that  number  which  will  divide  each  of 
them  without  a  remainder;  thus  5  is  the  common  meas- 
ure of  10,  20,  and  30;  and  the  greatest  number  that 
will  do  this,  is  called  the  greatest  common  measure. 

19.  What  is  meant  by  the  common  multiple? 

Ans.  Any  number  which  can  be  measured  by  two 
9r  more  numbers,  is  called  the  common  multiple  of 
.hose  numbers;  and  if  it  be  the  least  number  that  can 
:>e  so  measured,  it  is  called  the  least  common  multiple; 
hus  40,  60,  80,  100;  are  multiples  of  4  and  5;  but 
their  least  common  multiple  is  20, 

20.  When  is  a  fraction  said  to  be  in  its  lowest  terms? 
Ans,  When  it  is  expressed  by  the  smallest  numbers 

possible. 

21.  What  is  meant  by  a  prime  number? 

Ans.  It  is  a  number  which  can  only  be  measured 
by  itself,  or  an  unit. 

22.  What  is  meant  by  a  composite  number? 

Ans.  That  number,  which  is  produced  by  multiply- 
ing several  numbers  together,  is  called  a  composite 
number. 

23.  What  is  a  perfect  number? 

Ans.  A  perfect  number  is  one  that  is  equal  to  the 
sum  of  its  aliquot  parts.* 

*Tbe  following  perfect  numbers  are  all  which  are,  at  present  known 
6  8589869056 

28  137438691328 

496  2305843008139952128 

8128  24178521639228158837784575 

33550336          99035203 142S297 1830448816 128 


84  VULGAR  FRACTIONS. 

REDUCTION  OF  VULGAR  FRACTIONS. 

Reduction  of  Vulgar  Fractions,  is  the  bringing  of 
them  out  of  one  form  into  another,  in  order  to  prepare 
them  for  Addition,  Subtraction,  Multiplication,  &c. 

CASE  I. 
To  reduce  a  fraction  to  its  lowest  terms. 

RULE. 

Divide  the  greater  term  by  the  less,  and  that  divisor 
by  the  remainder,  and  so  continue  till  nothing  be  left; 
the  last  divisor  will  be  the  common  measure ;  then  divide 
both  parts  of  the  fraction  by  the  common  measure,  and 
the  quotients  will  express  the  fraction  required. 

Note. — If  the  common  measure  happen  to  be  1,  the 
fraction  is  already  at  its  lowest  term.  Cyphers,  on  the 
right  hand  side  of  both  terms,  may  be  rejected;  as-f  |£,  ^. 

EXAMPLES. 

1.  Reduce  iff  to  its  lowest  terms. 

144)240(1  48)144(3 

144  144 


96)144(1  i  Ans. 

96  48)240(5 

Greatest  common   —  240 

measure  48)36(2  

96 

Thus  48  is  the  greatest  common  measure,  and  the 
true  answer  is  obtained  by  dividing  the  fraction  by  it. 

This  reduction  may  be  performed,  also,  by  another 
rule,  thus : — Divide  the  numerator  and  denominator  of 
the  fraction  by  any  number  that  will  divide  them  both 
without  a  remainder ;  divide  the  quotients  in  the  same 
manner,  and  so  on,  till  no  number  will  divide  them 
both,  and  the  last  quotients  will  express  the  fraction  in 
its  lowest  terms. 

The  same  sum  done  by  this  method : — 
12)-ma2o         4)Jf(f  Answer. 

2.  Reduce  ¥9T  to  its  lowest  terms.  Ans.  *. 

3.  Reduce  iff  to  its  lowest  terms.  Ans.  * . 

4.  Reduce  fff- to  its  lowest  terms.  Ans.   ^. 

5.  Reduce  }        to  its  lowest  terms.  Ans.  I. 


REDUCTION  OF  VULGAR  FRACTIONS         85 

CASE  II. 

To  reduce  a  mixed  number  to  an  improper  fraction. 
RULE. 

Multiply  the  whole  number  by  the  denominator  of 
the  fraction,  and  add  the  numerator  to  the  product;  then 
set  that  sum,  namely,  the  whole  product,  above  the  de- 
nominator for  the  fraction  required. 

EXAMPLES. 

Reduce  23  f  to  an  improper  fraction. 
5 

117  47  Answer. 

2.  Reduce  12£  to  an  improper  fraction.       Ans.  iJ 

3.  Reduce  14T7T  to  an  improper  fraction.     Ans.  *T 

4.  Reduce  163^-  to  an  improper  fraction.  Ans.  3| 

CASE  III. 

To  reduce  an  improper  fraction  to  a  whole  or  mixed 

number. 

RULE. 

Divide  the  numerator  by  the  denominator,  and  the 
quotient  will  be  the  whole  or  mixed  number  sought. 

EXAMPLES. 

1.  Reduce  y  to  its  equivalent  number. 

3)12(4  Answer. 
12 

2.  Reduce  V  to  its  equivalent  number. 

7)15(2}  Answer. 
14 

1 

3.  Reduce  7ry  to  its  equivalent  number    Ans.  44T*T. 

5.  Reduce  y  to  its  equivalent  fraction.  Ans.  8. 

5.  Reduce  ^f3  to  its  equivalent  fraction.  Ans.  54|J. 

6.  Reduce  2|}8  to  its  equivalent  number.  Ans.  171||. 

CASE  IV. 

To  reduce  a  whole  number  to  an  equivalent  fraction, 
hating  a  ghtn  denojniniztor. 


Multiply  the  whole  numt  rr  by  the  r;ven   flenomina- 


36  REDUCTION  OF  VULGAR  FRACTIONS. 

tor ;  then  set  down  the  product  above  for  a  numerator, 
ind  the  given  denominator  below,  and  they  will  form 
the  traction  required. 

EXAMPLES. 

1.  Reduce  9  to  a  fraction  whose  denominator  shall 
be  7.  9X7=63,  then  6T3  is  the  answer. 

2.  Reduce  13  to  a  fraction  whose  denominator  shall 
be  12.  Ans.  iff. 

3.  Reduce  27  to  a  fraction  whose  denominator  shall 
bell.  Ans.Yr7- 

CASE  V. 
To  reduce  a  compound  fraction  to  a  simple  one. 

RULE. 

Multiply  all  the  numerators  together,  for  a  new  nu- 
merator, and  all  the  denominators  for  a  new  denomina 
tor,  then  reduce  the  fraction  to  its  lowest  term. 

EXAMPLES. 

1.  Reduce  *  of  !  of  I  to  a  single  or  simple  fraction. 

1X2X3         6         1 

-    _    -  =  -  =  -  Answer. 

2x3X4      24        4 

2.  Reduce  f  of  f  of  |f  to  a  single  fraction.  Ans.  /T 

3.  Reduce  ^  of  %  to  a  single  fraction.  Ans.'if. 

4.  Reduce  f  of  |  of  }J-  to  a  simple  fraction.  Ans.y/T. 

CASE  VI. 

To  reduce  fractions  of  different  denominations  to  oilier s 
of  the  same  value,  and  haying  a  common  denominator. 

RULE. 

Multiply  each  numerator  into  all  the  denominators 
except  its  own,  for  a  new  numerator,  and  all  th'e*  denom 
inators  into  each  other  for  a  common  denominator.* 


+The  least  common  denominator,  or  multiple,  of  two  or  more  numbers, 
may  be  found  thus:  Divide  tlie  given  denominations  by  any  number  that  will 
divide  two  or  more  of  them  without  a  remainder,  and  set  the  quotients  and 
undivided  numbers  underneath.  Divide  these  quotients  by  any  number  that 
will  divide  two  or  more  of  tliem  as  before,  and  thus  continue,  till  no  two 
numbers  are  left,  capable  of  bein?  lessened.  The.n  multiply  the  last  quotients 
and  the  divisor,  or  divisors  together,  and  the  product  will  be  the  answer. 

What  is  the  least  common  multiple  of  f  ,f  ,/y,  and  T^  ? 

8)9    8     15    16 


3)9    1     15      2 


3X1X5X2-30X3X8X720,  Ans. 


REDUCTION  OF  VLLGAK  Fit  ACTIONS.  87 

EXAMPLES. 

1.  Reduce  4,  5,  and  f  to  a  common  denominator. 

1  X3X4=12  the  numerator  for  *. 
2x2X4=16  the  numerator  for  3. 
3x2X3=18  the  numerator  for  I. 
Denominator  2x3x4=24  the  common  denominator. 
Therefore  the  results  are  £,  §,  !=£f  ,  |f  ,  ££. 

Or  the  multiplications  may  be   performed  mentally, 
and  the  results  given  1,  3,  !=££,  if,  if. 

2.  Reduce  f  and  £  to  a  common  denominator. 

Ans  J?  and 

3.  Reduce  §,  |,  and  I  to  a  common  denominator. 


5.  Reduce  4,  f  ,  and  f  to  fractions  of  a  common  de. 
nominator  Ans.  T4/^,  T\^  and  }$ 

CASE  VII. 

To  reduce  the  fraction  of  one  denomination  to  the  frac- 
tion of  another,  but  greater,  retaining  the  same  value. 

RULE. 

Make  the  fraction  a  compound  one,  by  comparing  it 
with  all  the  denominations  between  it  and  that  denom- 
ination to  which  you  would  reduce  it;  then  reduce  that 
compound  fraction  to  a  simple  one. 

EXAMPLES. 

1.  Reduce  |  of  a  cent  to  the  fraction  of  a  dollar.  By 
comparing  it,  it  becomes  -£  of  ^  of  yV  which  being  re- 
duced by  case  five,  will  be  4xlXl  =4  and  7x10x10 
=700.  Ans.  T|J.  D. 

2.  Reduce  f  of  a  mill  to  the  fraction  of  an  eagle. 


3.  Reduce  f  of  a  penny  to  the  fraction  of  a  pound. 
3X1  Xl=    3  1 

=£  - 


5x12x20=1200       400 

4.  Reduce  |  of  an  ounce  to  the  fraction  of  a  pound, 
Avoirdupois  Weight.  Ans.  -^Ib. 

5.  Reduce  J  of  a  dwt.  to  the  fraction  of  a  pound, 
Trow  Weight.  Ans.  T/¥7r  lb'. 

6.  Reduce  j  J  of  a  minute  to  the  fraction  of  a  (iny. 

i.     cla. 


BO  REDUCTION  OF  VULGAR  FRACTION*. 

CASE  VIII. 

To  reduce  the  fraction  of  one  denomination  to  the  frac- 
tion of  another,  but  less,  retaining  the  same  value. 

RULE. 

Multiply  the  given  numerator  by  the  parts  in  the 
denomination  between  it  and  that  to  which  you  would 
reduce  it,  and  place  the  product  over  the  given  denom- 
inator. 

EXAMPLES. 

1.  Reduce  T|T  of  a  dollar  to  the  fraction  of  a  cent. 
The  fraction  is  ^T  of  y  of  lj° ;  then 

i  xioxio   100     ,  A. .     ,     ,  . 

and  this  reueced,  is  equal  to 

175x  1  X  1      175  AnM 

2.  Reduce  IT-J¥¥  of  an  eagle  to  the  fraction  of  a  mill. 

Ans.  J. 

3.  Reduce  yJ-7  of  a  pound  to  the  fraction  of  a  penny. 

Ans.  1 

4.  Reduce  ^  of  a  pound  Avoirdupois,  to  the  fraction 
>f  an  ounce.  Ans.  '$-. 

5.  Reduce  r^T  of  a  pound  Troy,  to  the  fraction  of 
a  pennyweight.  Ans.  Jdwt. 

6.  Reduce  ,-jL^  of  a  day  to  the  fraction  of  a  minute 

Ans.  }f  of  a  min* 

CASE  IX. 

Tojind  the  value  of  the  fraction  in  the  known  part*  of 
the  integer;  or,  to  reduce  a  fraction  to  improper  value. 

RULE. 

Multiply  the  numerator  by  the  known  parts  of  the 
integer,  and  divide  by  the  denominator. 

EXAMPLES. 

1.  What  is  the  value  of  1  of  a  pound? 
2  thirds  of  a  nound. 
20 


3)40  thiroi  of  a  shilling. 
.  13  §.-|-t  third  of  ashiling. 

3)12  thirds  of  a  penny. 
~ 


Am.  13i.  4d 


REDUCTION  OF  VULGAR  FRACTIONS.  89 

2.  Reduce  &  of  a  shilling  to  its  proper  value. 

2  fifths  of  a  shilling. 
12 

5)24(4d. 
20 

4  fifths  of  a  penny, 
4 

6)16  fifths  of  a  farthing, 

3  qr.+l  fifth.  Ans.  4d.  3qr.  { 
3-  Reduce  f  of  a  Ib.  Troy,  to  its  proper  quantity. 

Ans.  7  oz.  4dwt 

4.  Reduce  £  of  a  mile  to  its  proper  quantity; 

Ans.  6  fur.  16  poles, 

5.  Reduce  ^  of  a  cwt.  to  its  proper  quantity, 

Ans.  2  qrs, 

6.  Reduce  f-  of  an  acre  to  its  proper  value. 

Ans.  2R.  20R 

7.  Reduce  ,^  of  a  day  to  its  proper  value. 

Ans.  7  hours  12  rain. 

r  CASE  X. 

To  reduce  any  given  quantity  to  the  fraction  of  a  great- 
er denomination  of  the  same  kind. 

RULE. 

Reduce  the  given  quantity  to  the  lowest  denomina- 
tion mentioned  for  a  numerator,  and  the  integer  into  the 
same  denomination  for  a  denominator. 

EXAMPLES. 

1.  Reduce  16s.  8d.  to  the  fraction  of  a  pound. 
16    8  Integer  £1 

12  2G 

Numerator     200  20 

==|  Ans.  12 

Denominator  240 

240  Denominator. 


IK)  ADDITION  OF  VULGAR  FRACTIONS. 

2.  Reduce  6  furlongs  and  16  poles  to  the  fraction  of 

a  mile  Ans.  f . 

3.  Reduce  |  of  a  farthing  to  the  fraction  of  a  pound. 


4.  Reduce  ^  dwt.  to  the  fraction  of  a  pound  Troy. 

Ans.  -3^ 

5,  Bring  80  cents  to  the  fraction  of  a  dollar. 

A  dollar  is  100  cents,  then  80  cents  are  equal  to  -J 
of  a  dollar;  which,  being  reduced,  is  equal  to  |  Ans, 
6)  Bring  16  cents  1)  mills  to  the  fraction  of  an  eagle. 
16  cents  9  mills  =  169    . 

: A  nc 

1  eagle  =  10000 

7.  Bring  2  quarters  3  J  nails  to  the  fraction  of  an  ell 
Fnglish.* 

2  quarters  3i  nails. 
4 

11 
9 

Numerator   100 

Denominator   9  of  4-  of  1  =  *-$%  =  4  Ans. 


ADDITION  OF  VULGAR  FRACTIONS. 
CASE  I. 

To  add  fractions  having  a  common  denominator. 

RULE. 

Add  all  the  numerators  together,  and  place  the  sum 
over  the  common  denominator,  which  will  give  the  sum 
of  the  fractions  required. 

EXAMPLES. 

1.  Add  I,  A  and  £  together. 
______       iXf  Xi=f =1*  Answer. 


*When  the  sum  contains  a  fraction,  as  in  the  seventh  example, 
multiply  both  parts  of  the  sum  by  the  denominator  thereof,  and  to 
the  numerator  add  the  numerator  of  the  given  fraction. 


ADDITION  Oi'  VULUAIl  FRACTION'S.  91 

2.  Add-^,2,?-  and  £  togctlier. 

'  -}~M- +•?•+•?-  =  V  =  H  Answer. 

CASE  II. 
To  add  fractions  having  different  denominators. 

RULE. 

Find  the  common  denominator  by   Case  VI.  in  Re- 
duction; then  add,  as  in  the  preceding  examples. 
.    EXAMPLES. 

1.  Add  J  and  £  together. 

4X5=20> 
3X9=27$ 

47  sum. 
4x^=36  com.  denom.  ££  =  !££  Ans. 

2.  Add  f  and  -fa  together.  Ans.  T9T. 

CASE  III. 
To  add  mixed  numbers. 

RULE. 

Add  the  fractions  as  in  Case  I,  in  Addition,  and  the 
whole  numbers  as  in  Simple  Addition;  then  add  the 
fractions  to  the  sum  of  the  whole  numbers.  .  If  the 
fractions  have  different  denominators,  reduce  them  to  a 
common  denominator,  and  then  add  the  fractions  to  the 
integers  or  whole  numbers. 

EXAMPLES. 

1.  Add  13^,  9T4T,  3^  together. 

13+9+3=25  whole  numbers. 
TV+T45+777=if =f      Thus,  25|  Ans. 

2.  Add  5f ,  6|  and  4i  together. 

5-|_6-|-4=15  whole  numbers. 
Then,  2x^X2=32 
7x3x2=42 


98  sum  of  the  numerators. 

3x8X2=48  common  denominator. 
Then,  ff=2^¥.          Thus,  15+2^= 17 Jj  Answer. 

3.  Add  If,  2|  and  3-f  togetner.  Ans.  7}£J 


~  ADDITION  OF  VULGAR  FRACTIONS. 

CASE  IV. 
To  add  compound  fractions. 

RLLE. 

Reduce  them  to  simple  ones,  and  proceed  as  before. 

EXAMPLES. 

1.  Add  i  of  |  of  |,  to  f  of  f  of  |f. 

*-^*  simple  fraction* 


3  =  •      !    - 

18X5X11=®^  simp    fractlon- 
Then  find  a  common  denominator. 

4v  4  _  \Q 
forj,  T4rtnusiCn-  -11  numerator. 

27  sum  of  the  numerators. 

4x11=44  common  denominator. 

Therefore  JJ  is  the  answer. 

2.  Add  J  of  J,  and  £  of  i£  together.  Ans. 

3.  Add  |  of  -j^,  and  T^  of  f  together.  Ans. 

4.  Add  f  ,  9-}  and  |  of  1  together. 

Note.  —  The  mixed  number  of  9j  =  -y5  ;  the  compound 
fraction  |  of  i=|.  Then  the  fractions  are,  ^,  y  and 
|>  which  must  be  'reduced  to  the  fractions  of  a  common 
denominator  and  added.  Ans. 

5.  Add  1^,  6|,  |  of  J  and  7j  together.  Ans. 

CASE  V. 
When  the  given  fractions  are  of  several  denominations. 

RULE. 

Reduce  them  to  their  proper  values,  or  quantities, 
and  add  them  according  to  the  following  examples. 

EXAMPLES. 

1.  Add  S-  of  a  pound  to  J  of  a  shilling. 

Thus,  |  of  a  pound=13s.     4d. 

and  f  of  a  shilling=  Os.     4d.     3iqr. 

13s.     8d.     3lqr.  Ans. 

2.  Add  I  of  a  pound  and  T3F  of  a  shilling  together. 

Ans.  15s.  lO^d. 

3.  Add  £  of  a  week,  £  of  a  day,  and  1  of  an  hour 
together.  Ans.  2d. 


SUBTRACTION  OF  VULGAR  FRACTIONS. 

4.  Add  I  of  a  yard,  -J  of  a  foot,  and  £  of  a  mile  to- 
gether.  Ans.  llGOyds.  2ft.  7iru 

5.  Add  £  of  a  dollar,  f  of  a  cent,  -f^  of  a  cent,  and 
of  a  mill  together.  Ans.  20c.  9m. 

6.  Add  -J-  of  a  pound,  ^  of  a  shilling,  and  «  of  a  pen- 
ny together.  Ans.  2s.  ~ 


SUBTRACTION  OF  VULGAR  FRACTIONS. 
CASE  I. 

When  the  fractions  have  a  common  denominator. 

RULE. 

Subtract  the  less  numerator  from  the  greater,  and  set 
the  remainder  over  the  common  denominator,  which 
will  show  the  difference  of  the  given  fractions. 

EXAMPLES. 

1.  Subtract  f  from  4.  Ans.  -f . 

2.  What  is  the  difference  between  |  and  I? 

Ans.  |=r 

2.  Take  ^  from  T^.  Ans.  T2?=; 

4.  Take  f  from  f  Ans.  |=i. 

CASE  II. 

When  fractions,  or  mired  numbers,  are  to  be  subtracted 
from  whole  numbers. 

RULE. 

Subtract  the  numerator  from  its  denominator,  and 
under  the  remainder  place  the  denominator;  then  carry 
one  to  be  deducted  from  the  whole  number. 

EXAMPLES. 

1.  Take  f  from  12. 

Thus,    12. 


llf  Answer. 

2.  Subtract  27f}  from  32.  Ans.  4 

3.  From  10,  take  TV.  Ans.  9 

4.  From  9,  take  5f  Ans. 

5.  From  25,  take  24^.  Ans.  TV 


94  SUBTRACTION  OF  VULGAR  FRACTIONS. 

CASE  VI. 
To  subtract  fractions  hating  different  denominators. 

RULE. 

Reduce  the  fractions  to  a  common  denominator,  by 
Ca^o  VI.  in  Reduction,  and  subtract  the  less  numerator 
from  the  greater — the  difference  will  be  the  answer. 

EXAMPLES. 

1.  What  is  the  difference  between  ij  and  JJ? 

Thus  If  and  } f  are  equal  to  |JJ,  #ff, 

And  88  from  171,  leaves  83.  Ans.  7\3 

2.  From  f  take  f .  Ans. 

3.  Take  £  from  f .  Ans. 

4.  Subtract  -fe  from  ^-.  Ans.  J 

CASE  VI. 

To  distinguish  the  largest  of  any  two  fractions. 

RULE. 

Reduce  them  to  a  common  denominator,  and  the  one 
that  has  the  larger  numerator  is  the  larger  fraction. 

EXAMPLE. 

Which  is  the  greater  fraction,  j-J  or  ||? 
Thus  192  common  denominator. 
12x15=180  numerator. 
=  176  numerator. 

4  numerator 
=A. 
Therefore  -J-f  is  the  greater  fraction  by  J^,  Ans. 

CASE  V. 

To  subtract  one  mixed  number  from  another,  when  the 

fraction  to  be  subtracted  is  greater  than  that 

from  which  the  subtraction  is  to  be  made. 

RULE. 

Reduce  the  fractions  to  a  common  denominator;  sub- 
tract the  numerator  of  the  greater  from  the  common 
denominator,  and  add  to  the  remainder  the  less  numera- 
tor; then  set  the  sum  of  them  over  the  common  denom- 
inator, and  carry  one  to  the  whole  number,  and  sub- 
tract as  in  Simple  Subtraction. 


MULTIPLICATION  OF  VULGAR  FRACTIONS.  95 

EXAMPLES. 

1.  From  12 J    subtract  8}£, 

Thus  |  reduced  to  a  common  denominator ,=  JJ~, 
and  }|.  reduced  to  a  common  denomipator,=^T7A. 

Then  72  taken  from  114,  leaves  42;  which,  added  to 
•7,  the  less  numerator,  makes  99  for  the  numerator  in 
he  answer.  Then  carrying  1  to  the  whole  number, 
namely,  8,  makes  it  9;  and  taking  9  from  12  leaves  3. 

Therefore  the  answer  is 
2    From  10T\,  take  1TV  Ans. 

CASE  VI. 
When  fractions  are  of  different  denominations. 

RULE. 

Reduce  them  to  their  proper  values,  or  quantities^ 
and  subtract  as  in  Compound  Subtraction. 

EXAMPLES. 

1.  From  fof  a  pound,  take  1  of  a  shilling. 

Thus,  f  of  a  pound  =17s.     6d. 
And  i  of  a  shilling  =0       4 

17s.    2d.  Answer. 

2.  From  £  of  a  ton  take  /ff  of  a  cwt. 

Ans.  14cwt.  Oqr.  lllb.  3oz.  3£dr. 

3.  From  J  of  a  pound,  take  J  of  a  shilling  and  what 
will  be  the  remainder?  Ans.  14s.  3d. 

4.  From  £  of  a  pound,  Troy  Weight,  take  £  of  an 
ounce.  Ans.  8oz.  16dwt.  16gr. 


MULTIPLICATION  OF  VULGAR  FRACTIONS. 

RULE. 

Reduce  compound  fractions  to  simple  ones,  ami  mix 
ed  numbers  to  equivalent  fractions;  then  multiply  all 
the  numerators  together  for  a  numerator,  and  all  the 
denominators  together  for  a  denominator  which  will 
give  the  product  required. 


>  MULTIPLICATION  OF  VULGAR  FRACTIONS. 

EXAMPLES. 

1.  Multiply  I  by  f. 

Here,  J  X§=3%==o  the  answer. 

2.  Multiply  f  by  }.  Ans.  ^ 

4.  "  J*of7by|.  Ans.  1}. 

5.  "  t>J  by  4-.  Ans.  1 

6.  «  4££  by  3-5.  Ans.  14±< 

7.  "  ^ibyi  Ans-^ 


DIVISION  OF  VULGAR  FRACTIONS. 

RULE. 

Reduce  compound  fractions  to  simple  ones,  and  mix- 
ed numbers  to  equivalent  fractions;  thea  multiply  the 
numerator  of  the  dividend  by  the  denominator  of  the 
divisor,  for  a  new  numerator,  and  the  denominator  ol 
he  dividend  by  the  numerator  of  the  divisor,  for  the  de- 
nominator; the  fractions  thus  formed  will  be  the  answer. 

EXAMPLES. 

1.  Divide  4  by  J. 

Thus,  4  numerator  of  the  dividend, 
3  X  denominator  of  the  divisor. 

12  numerator. 

Then  7  denominator  of  the  dividend. 
2  X  numerator  of  the  divisor. 

11  denominator 

Therefore,   J-J=-J  is  the  answer 

2.  Divide  %  by  }. 


. 
g3assg  Answer. 

3.  Divide  U  by  .*.  Ans.  4 

4.  "      T\byf.  Ans. 

5.  «      I  by  Vs-  Ans. 

6.  «      ^J  by  f  .  Ans. 

7.  •«      %  by  ->.  Ans. 

8.  <4      ^  by  ^.  Ans.  T£ 


DECIMAL  FRACTIONS.  97 

9,  Divide  f  by  2.  Ans.  TV 

10,        «      7|by9f.  Ans.  fj. 

11-        "      I  of  £  bY  4  of  7f  •  Ans-  rfr- 

12,  What  part  of  33  JT,  is  26|i.?  Ans.  f 

Questions. 

1.  What  are  Vulgar  Fractions? 

2.  How  are  they  represented  in  figures  ? 

3.  What  is  the  upper  figure  called? 

4.  What  is  the  lower  figure  called  ? 

5.  What  does  the  denominator  show  ? 

6.  What  does  the  numerator  show  ? 

7.  What  are  the  two  numbers  cf  a  fraction  some- 

times called  ? 


DECIMAL  FRACTIONS. 

Decimal  Fractions  are  parts  of  whole  numbers,  and 
nre  separated  from  them  by  a  point,  thus,  8.5;  which  is 
read,  eight  and  five  tenths,  or  8T5^.  All  the  figures  on 
the  left  of  the  point  are  whole  numbers;  those  on  the 
right  arc  fractions.  An  unit  is  supposed  to  be  divided 
into  ten  equal  parts,  and  the  figure  at  the  right  of  the 
point  expresses  the  number  of  those  parts.  Decimals 
decrease  in  a  tenfold  proportion,  as  they  depart  from 
the  separating  point.  Thus,  .5  is  5  tenths,  or  one  half; 
.57  is  57  hundredths;  .05  is  5  hundredths;  and  .005  is  5 
thousandths.  Cyphers  placed  at  the  right  hand  of  de- 
cimals do  not  alter  their  value;  thus,  .5  or  T\;  .50  or 
T$$ ;  .500  or  1  £  J£ ,  are  all  of  the  same  value,  and  equal 
to  £.  The  first  place  of  decimals  is  called  tenths ;  the 
second,  hundredths,  &c. 

DECIMAL  NUMERATION  TABLE. 

07 

o?       v>  "^       tr> 

&     "^  •     r3       3 

J  S   „•  £  £  3  J    . 

2     T3      en  "is    "-*      w     "**      w 

ill  fj    .  ^ i  ill  1 

i 1  I  J 1  §  I    Illgll 

7654321.654321 


98  ADDITION  OF  DECIMALS. 

ADDITION  OF  DECIMALS. 

RULE. 

Place  the  figures  according  to  their  values — units  un- 
der units,  tenths  under  tenths,  &,c.  and  add  as  in  Sim- 
ple Addition  of  whole  numbers;  observing  to  place  the 
point  in  the  sum  under  those  in  the  given  numbers. 

EXAMPLES. 

1.  Add  together  the  following  sums,  viz:  252.25. 
343.5,  17.85,  1244.75  and  .425. 

Thus,    252.25         Note.— The  answer  to  this  sum 
343.5       is  read  thus:  One  thousand  eight 
17.85     hundred  and  fifty  eight,  and  seven 
1244.75     hundred    and    seventy-five    thou- 
.425  sandths. 


1898.775  Answer. 

ii.  in. 

87654.321  987654.3 

23456.78  212345.67 

98765.4  898765.432 


209876.501  2098765.402 


4.  Add  420.4,  38.05,  54.9,  27.003  and  29.384. 

Ans.  569.737. 

5.  Add  376.25,  86.125,  6.5,  41.02  and  358-865. 

An?.  868.760. 

6.  Add  .64,  .840,  .4,  .04,  .742,  .86,  .99  and  .450.   I 

Ans.  4.9H2.: 

7.  William  expended  for  a  gig  $255^,  for  a  wagon 
7-flfr,  fora  bridle  Ty¥  and  for  a   saddle    '$ilifffc; 

What  did  they  amount  to?  Ans$804.455. 


JVbfe. — Dimes,  cents  and  mills  are  decimals  of  a  dollar.  A  dime 
is  one  tenth,  a  cent  is  one  hundredth,  a  mill  is  one  thousandth;  which 
shows  that  the  addition  of  Federal  Money  is  the  addition  of  decimals. 
Thus  5  tenths  of  a  dollar  is  the  same  as  oO  hundredths,  or  50  cents; 
and  25  hundredths  of  a  dollar  is  equal  to  25  cents,  fee.  It  may  be 
likewise  added,  that  .5,  or  .50,  or  .500  being-  equal  to  one  half,  .25 
equal  to  one  quarter,  and  .75  equal  to  three  quarters  or  three  fourths, 
»'O  .7,  or  .35,  or  any  intermediate,  fractious,  have  a  proportionate 


SUBTRACTION  OF  DECIMALS. 

8.  James  bought  2/>  cwt,  of  sugar,  23.265  cwt.  of 
hay  and  4,2657  of  rice.     How  much  did  he  buy  in  all  ? 

Ans.  30,0307cwt. 

9,  James  is  14^  years  old,  John  15-^-  and  Thomas 
16T7^.     What  is  the  sum  of  all  their  ages? 

Ans.  46.5  years. 

xO.  What  is  the  sum  of  one  and  five  tenths;  forty-five 
and  three  hundred  and  forty  nine  thousandths;  and  six- 
teen hundredth*?  Ans,  47.009. 


SUBTRACTION  OF  DECIMALS. 

RULE. 

Write  the  larger  number  first,  and  the  smaller  one 
under  it;  then  subtract  as  in  Simple  Subtraction;  obser- 
ving, that  the  dividing  point  in  the  answer,  or  remain- 
der, must  be  placed  under  those  in  the  sum. 

EXAMPLES. 
I.  II.  III. 

From  91.73  2.73  1.5 

take    2.138  1.9185  .987654321 


89.592  0.8115  0.512345679 


4.  Bought  a  hogshead  of  molasses,  containing  60.72 
gallons  and  sold  40.721   gallons.     How  much  was  left 
in  the  hogshead?  Ans.  19.999  galls. 

5.  A  merchant,  owing  $270.42,  paid  $  191.626;  how 
much  did  he  then  owe?  Ans.  78.794. 

6.  I  bought  20,25  yards  of  cloth  and  sold  5.75  yards. 
How  much  had  I  left?  Ans.  14.50  yds. 

7.  From  .650  of  a  barrel  take  .6  of  a  barrel. 

Ans.  .050  barrel. 

8.  From  •£$£§  of  a  bushel  take  -J^  of  a  bushel. 

Ans.  .126  bushel. 

9.  A  farmer  was  42.075  years  old  when  he  came  to 
the  Western  country,  and  is  now  64.67  years  of  age. 
How  long  since  he  emigrated?  Ans.  22.595  years. 

10.  Take  one  hundredth  from  on  tenth.         Ans.  .09. 


100     MULTIPLICATION  AND  DIVISION  OF  DECIMALS. 

MULTIPLICATION  OF  DECIMALS. 
RULE. 

Place  the  multiplier  under  the  multiplicand,  and  mul- 
tiply as  in  Simple  Multiplication;  then  point  off  as  ma- 
ny places  for  decimals  as  there  are  decimals  in  the 
multiplicand  and  multiplier.  If  there  be  not  so  many 
figures  in  the  product  as  there  are  decimals  in  both  fac- 
tors, the  deficiency  must  be  supplied  by  prefixing  cy- 
phers. 

EXAMPLES. 

in. 

.63478 
.8994 

253912 
571302 
571302 
507824 

.570921132 

4.  Multiply  .63478  by  .8204.         Ans.  .520772512. 

5.  Multiply  .385746  by  .00464,  Ans.  00178986144. 

6.  What  will  5.66  bushels  of  wheat  cost  at  $1.08  a 
bushel?  Ans.  $6.1128,  or  $6.11c.  2Tyn. 

7.  What  will  8,6  pounds  of  flour  come  to  al  $.04  a 
pound?  Ans.  $  .34cts,  4m. 

8.  At  $  .25c.  a  bushel,  what  will  12.67  bushels  of 
apples  cost?  Ans,  3.1675. 

9.  If  I  travel  30.75  miles  a  day,  how  far  shall  1 
travel  in  8.325  days  ?  Ans.  255,99375  miles. 


Multiply  24.85 
by    6.25 

12425 
4970 
14910 


228375 
319725 
365400 


155.3125       3996.5625 


DIVISION  OF  DECIMALS. 

RULE. 

Divide  as  in  Simple  Division,  and  point  off  as  many 
figures  from  the  right  hand  of  the  quotient,  for  decim- 
als, as  the  decimal  figures  in  the  dividend  exceed  in 
number  those  in  the  divisor.  When  there  are  not  so 
many  figures  in  the  quotient  as  this  rule  requires,  the 


DIVISION  OF  DECIMALS. 


101 


deficiency  must  be  supplied  by  prefixing  cyphers  to  the 
left  of  the  quotient.  When  there  are  more  decimal  fig- 
ures in  the  divisor  than  in  the  dividend,  place  as  many 
cyphers  to  the  right  of  the  dividend  as  will  make  them 
equal. — When  the  number  of  decimals  in  the  divisor, 
and  the  number  in  the  dividend  are  equal,  the  quotient 
will  always  be  in  whole  numbers,  unless  there  should 
be  a  remainder  after  the  dividend  is  all  brought  down. 
When  there  is  a  remainder,  cyphers  must  be  annexed 
to  it  and  the  division  continued  and  the  quotient  thence 
arising  will  be  decimals. 

EXAMPLES. 
I.  II. 

.5).75(1.5  *"   324.8)9876.5(30.4079 
5  9744 

25  13250  cypher 

25  12992  annexed. 


IV. 

6.4)128.64(20.1 
128 

64 
64 


25800 
22736 

30640 
29232 

1408+ 


in. 

.48)65.88(137 

48 

178 
144 

348 
336 

12  rem, 


5.  Divide  234.70525  by  64.25.  Ans.  3,653. 

6.  Divide  14  by  .7854.  Ans.  17.825. 

7.  Divide  2175.68  by  100.  Ans.  21.7568. 

8.  If  you  divide  116.5  barrels  of  flour  equally  among 
5  men,  how  many  barrels  will  each  have? 

Ans.  23.3  barrels. 

9.  At  $  .25  a  bushel,  how  many  bushels  of  corn 
may  be  bought  for  $300.50.  Ans.  1202  bushels. 

10.  At  $  .12£  or  $  .125  a  yard,  how  many  yards  of 
cloth  may  be  bought  for  $16?  Ans.  128  yds. 

11.  Bought  128  yards  of  tape,  for  $•  .64;  how  much 
was  that  a  yard?  Ans.  $  .005,  or  5  mills. 


102  REDUCTION  OF  DECIMALS. 

REDUCTION  OF  DECIMALS. 
CASE  I. 

To  reduce  a  vulgar  fraction  to  a  decimal. 

RULE. 

Place  cyphers  to  the  right  of  the  numerator  until 
you  can  divide  it  by  the  denominator;  and  divide  till 
nothing  remains;  or,  if  it  be  a  number  th^t  will  not  di- 
vide without  a  remainder,  then  divide  until  you  get 
three  or  more  figures  for  the  quotient.  The  quotient 
will  be  the  vulgar  fraction  expressed  in  decimals. 

EXAMPLES, 

1.  Reduce  1  to  a  decimal. 

Thus,  2)1.0(3 
10 

2»  Reduce  }  and  f  to  decimals. 

4)1.00(55  4)3.00(.75 

8  28 


20  20 

20  20 

3.  Reduce  £  to  a  decimal.  Ans.  .375. 

4.  Change  j,  f ,  -fe  an<*  TJ  to  decimals. 

Ans.  .125,  .5,  .75.  04, 

5.  What  decimal  is  equal  to  ^1  Ans.  .05.     What  is 
equal  to  J?  Ans.  .2.     What  is  equal  to  £?  Ans.  .3333-f. 

6.  Change  64T3¥  bushels  to  its  equivalent  value. 

Ans.  64.25  bushels. 

CASE  II. 

To  reduce  any  sum,  or  quantity,  to  the  decimal  of  any 
given  denomination. 

RULE. 

Reduce  the  quantity  to  the  lowest  denomination,  and 
reduce  the  proposed  integer  to  the  same  denomination 
then  divide  the  quantity  by  the  amount  of  the  integer 
and  the  quotient  will  be  the  answer. 


REDUCTION  OF  DECIMALS.  103 

EXAMPLES. 

1.  Reduce  3s.  9d.  to  the  fraction  of  a  pound. 
One^pound  reduced  to  pence  makes  240;  and  3s.  9d 
reduced  to  pence  makes  45. 

Then,  240)45.0000(.1875  Answer. 
240 

2100 
1920 


1800 
1680 

1200 
1200 

The  same  sum  may  be  done  by  writing  the  given 
numbers  from  the  least  to  the  greatest  in  a  perpendicu- 
lar column,  and  dividing  each  of  them  by  such  number 
as  will  reduce  it  to  the  next  denomination,  annexing 
the  quotient  to  the  succeeding  number. 


Thus,  12 


9.00 


2|0  3.750|0 

.1875  Answer. 


2.  Reduce  7  drams  to  the  decimal  of  a  pound,  Avoir- 
dupois Weight.  Ans.  .02734375. 

3.  Reduce  14  minutes  to  the  decimal  of  a  day. 

Ans.  .009722. 

4.  Reduce  21  pints  to  the  decimal  of  a  peck. 

Ans.     1.3125. 

5.  Reduce  15s.  6d.  to  the  decimal  of  a  pound. 

Ans.  .775. 

6.  Reduce  56  gallons  3  quarts  1  pint  to  the  decimal 
of  a  hogshead.  Ans.  .9027. 

7.  Reduce  12  dwts.  16grs.  to  the  decimal  of  a  pound 
Troy  Weight.  Ans.  .0527, 

8.  Reduce  4  mills  to  the  decimal  of  a  dollar.    . 

Ans.  .004 


.-— In  doing  sums  in  this  rule,  it  will  be  necessary  to  keep  in 
mind  the  tables  of  the  different  weights,  measures,  money,  &c. 


104  REDUCTION  OF  DECIMALS. 

CASE  III. 

To  find  the  value  of  any  decimal  fraction. 

RULE. 

Multiply  the  decimal  by  the  number  of  parts  in  the 
next  lower  denomination ;  point  off  as  many  figures  for 
decimals  as  is  required  by  the  rule  in  multiplication  of 
decimals;  then  multiply  the  decimal  by  the  number  of 
parts  in  the  next  lower  denomination,  and  so  on,  to  the 
last.  The  figures  on  the  left  of  the  points  will  show  the 
value  of  the  decimal  in  the  different  denominations. 

EXAMPLES. 

1.  What  is  the  value  of  .775  of  a  pound? 
£.775 
20 


^.15.500 
12 

d.6.000  Answer. 


2.  What  is  the  value  of  .625  of  a  cwt.? 
4 


2.500 

28 

4000 
1000 

14.000  Ans.  2  qr.  1416. 

3.  What  is  the  value  of  .625  of  a  shilling?  Ans.  7 

4.  What  is  the  value  of  .4694  of  a  pound,  Troy 
Weight?  Ans.  5oz.  12dwts.  15.744grs 

5.  What  is  the  value  of  .6875  of  a  yard? 

Ans.  2qrs.  3na. 

6.  What  is  the  value  of  .3375  of  an  acre? 

Ans.  1R.  14P, 

7.  What  is  the  value  of  .0008  of  an  Eagle? 

Ans.  8m 


DUODECIMALS.  105 

Questions. 

1.  What  are  decimal  Fractions? 

2.  How  are  they  separated  from  whole  numbers? 

3.  In  what  manner  do  they  decrease  as  they  depart 

from  the  separating  point? 

4.  In  the  table  of  numeration,  what  is  the  first  place 

called  ? 

5.  What  money,  or  currency,  is  reckoned  after  the 

manner  of  Decimal  Fractions  ? 


DUODECIMALS. 

Duodecimals  are  fractions  of  a  foot  or  of  an  inch,  or 
parts  of  an  inch,  and  have  12  for  their  denominator. — 
They  are  useful  in  measuring  planes,  or  surfaces,  and 
solids.  In  adding,  subtracting,  and  multiplying  by  Du 
odecimals,  it  is  necessary  to  carry  one  for  twelve. 

The  denominations  are  feet,  inches,  seconds,  thirds 
and  fourths. 

12  fourths""         make         1  third 

12  thirds      -  -       1  second      ". 

12  seconds  -  -       1  inch          /. 

12  inches  1  foot          Ft. 


MULTIPLICATION  OF  DUODECI3IALS. 

RULE. 

Set  down  the  different  denominations,  one  under  the 
other,  so  that  feet  stand  under  feet,  inches  under  inches, 
seconds  under  seconds,  &/c.  Multiply  each  denomina 
tion  in  the  sum,  by  the  feet  in  the  multiplier,  and  set 
the  result  of  each  under  its  corresponding  term,observ 
ing  to  carry  one  for  every  12  from  one  denomination 
to  another.  Then  multiply  the  sum  by  the  inches  in 
the  multiplier,  and  set  the  result  of  each  term  one  place 
removed  to  the  right  of  those  in  the  sum ;  and  in  like 
manner,  multiply  the  sum  or  multiplicand  by  seconds, 
thirds,  &,c.  if  there  be  any  in  the  multiplier. 

Or,  instead  of  multiplying  by  inches,  &c.  take  such 
parts  in  the  multiplicand,  as  these  are  of  a  foot. 


106  DUODECIMALS. 

Add  the  amount  of  the  multiplications  together,  and 
their  sum  will  be  the  answer. 

EXAMPLES. 

i.  IT. 

Ft.  L  Ft.     7. 

Multiply  47  14     9 

by  6     4  .46 


59     0 
746 


29    0    4  66     4     6 


.\otc  1. — Feet  multiplied  by  feet,  give  feet.  Feet  mul- 
tiplied by  inches  give  inches.  Feet  multiplied  by  se- 
conds, give  seconds.  Inches  multiplied  by  inches,  give 
seconds.  Inches  multiplied  by  seconds,  give  thirds. 
Seconds  multiplied  by  seconds,  give  fourths. 

III.  IV. 

Ft.    /.      "     "'  Ft.    /. 

Multiply  8      4      2     10  11     10 

by  4      2  10       9 


33       4     11       4  118       4 

14858  8     10     6 


34       9       7       9     8""          127       2     6" 


Note  2. — 'In  doing  the  third  sum,  I  begin  with  4, 
which  stands  under  the  8,  and  multiply  the  sum,  begin- 
ning with  the  right  hand  figure  which  is  10;  saying  4 
limes  10  are  40.  In  40, 1  find  there  are  3  times  12  and 
4  over.  Setting  down  4,  I  multiply  the  next  figure, 
adding  three  to  it,  which  makes  11.  and  thus  multiply 
the  whole  sum.  Then  taking  the  2  for  the  multiplier, 
I  say  2  times  ten  are  20.  In  20  1  find  12  is  contained 
once,  and  8  over.  Setting  down  8  one  place  farther  to 
the  right,  I  say  2  times  2  are  4,  and  one  to  carry  makes 
5;  and  after  this  manner  multiply  all  the  figures  in  the 
sum.  Then  adding  the  two  rows  of  figures  together,  I 
obtain  the  answer 


DUODECIMALS. 


107 


Method  of  doing  the  same  sum  by  taking  the  frac- 
tional parts. 

Ft.      7.      "    "' 


2  inches = 


842     10 
4       2 


33 
1 


4     11 
4      8 


4 

5    8 


34       9 


9     8""  Answer* 


In  this  last  example,  I  multiply  the  sum  by  4,  as  in 
the  former  case.  Then,  as  2  inches  make  -J  of  a  foot,  I 
divide  the  sum  by  6,  which  I  hnd  multiplied  by  4,  divi- 
ding it  after  the  manner  of  Compound  Division,  multi- 
plying each  remainder  by  12,  and  adding  it  to  the  next 
lower  denomination;  and  setting  the  result  under  the 
amount  of  the  multiplication.  Then  1  add  the  two 
sums  as  before. 

I.  What  jrre  the  solid  contents  of  a  cubick  block  that 
is  4  feet  4  inches  in  length,  3  feet  8  inches  in  breadth, 
and  2  feet  8  inches  in  thickness? 

Ans.  42JFV.  41.  5"  4"'. 

6.  What  i?  the  product  of  12  feet  9  inches,  multiplied 
by  6  feet  4  inches.  Ans.  SO  Ft.  97. 

7.  What  is  the  product  of  3  feet  2  inches  3"  multi- 
plied by  3  feet  2  inches  3"? 

Ans.  10.FUJ.  11".  0'".  9"". 

8.  What  is  the  price  of  a  marble  slab    whose  length 
is  5  feet  7  inches,  and  breadth  1  foot  10  inches,  at  one 
dollar  per  foot?  Ans.  $10.23. 

9.  How  rnnnv  square  feet  in  a  board   177V.  77.  long 
and  IFl.  57.  wide?  Ans.  24F*.  107.  II". 

10,  How  many  solid  feet  in  a   load   of  wood  6Ft.  77. 
long  and  3R  57.  high  and  3JFV.  87.  wide? 

Ans.  S2Ft.  57.  8".  4"', 

11.  What  will  be  thr,  expense  of  plastering  the  walls 
of  a  room  S.f-1.  67.  high  arid  each  of  the  4  sides    IfiFt. 
37.  long,  at  50  cents  per  square  yard ?     Ans.  $30 .(590-}-. 


108  SINGLE  RULE  OF  THREE. 

Questions. 

1.  What  are  Duodecimals? 

2.  In  what  are  they  useful  ? 

3.  In  adding,  subtracting,  and  multiplying  Duodeci- 

mals, what  do  you  observe  in  carrying  from  one 
denomination  to  another? 

4.  What  are  the  denominations  used  in  Duodecimals? 

5.  Repeat  the  rule  for  Multiplication  of  Duodecimals? 


SINGLE  RULE  OF  THREE. 

The  Rule  of  Three,  which  is  sometimes  called  the 
Rule  of  Proportion,  teaches  how  to  find  a  fourth  pro- 
portional to  three  numbers  given.  As  it  has  three  terms 
given  to  find  a  fourth,  it  is  generally  called  the  Rule  of 
Three. 

Questions  to  prepare  the  learner  for  this  rule. 

1.  If  2  apples  cost  3  cents,  how  much  will  4  apples 
cost  at  the  same  rate  ? 

2.  If  you  give  2  cents  for  4  nuts,  how  many  cents 
must  you  give  for  8  nuts? 

3.  If  a  pound  of  butter  cost  8  cents,  how  much  will 
pounds  cost? 

4.  A  boy  has  20  melons  to  sell,  and  asks  10  cents  for 
two,  how  much  will  they  all  come  to  at  the  same   rate? 

5.  If  6  men  can  reap  a  field  of  wheat  in  4  days,  how 
long  will  it  take  12  men  to  reap  the  same  field? 

6.  If  4  yards  of  cloth  cost  1  dollar,  how  much  will  2 
yards  cost? 

7.  How  much  will  a  gallon  of  milk  come  to,  at  four 
cents  a  quart? 

8.  How  much  will  a  bushel  of  peaches  come  to  at  25 
cents  a  peck  ? 

9.  If  3  cents  will  buy  2  apples,  how   many  apples 
will  9  cents  buy? 

10.  If  a  boy  can  run  2  miles  in  one  hour,  how  far  can 
e  run  in  4  hours? 


SINGLE  RULE  OF  THREE.  109 

RULE. 

Set  the  term  in  the  first  place,  which  is  of  the  same 
kind  with  that  in  which  the  answer  is  required.  Then 
determine  whether  the  answer  ought  to  be  greater  or 
less  than  the  third  term.  If  the  answer  ought  to  be 
greater  than  the  third  term,  set  the  greater  of  the  other 
two  numbers  on  the  left  for  a  second  or  middle  term; 
and  'hs  L  ss  number  on  the  left  of  the  second  term,  for 
a  first  term.  If  the  answer  ought  to  be  less  than  the 
third  term,  the  less  of  the  two  other  numbers  must  be 
the  mi. Idle  t2rm,  and  the  greater  must  be  the  first  term. 

After  iluis  stating  the  sum,  proceed  to  do  it  in  the  fol- 
lowing manner,  viz:  Reduce  the  third  term  to  the  lowest 
denomination  mentioned  in  it.  Reduce,  likewise,  the 
first  and  second  terms  to  the  lowest  denomination  that 
either  of  them  has.  Then  multiply  the  second  and 
third  terms  together,  and  divide  their  product  by  the 
first  term.  The  quotient  thus  obtained  will  be  the 
answer. 

It  will  not  be  necessary  to  distinguish  between  direct 
and  inverse  proportion,  because  the  foregoing  rule  is 
calculated  for  both. 

PROOF. 
By  reversing  the  statement. 

EXAMPLES. 

1.  If  3  pounds  of  sugar  cost  25  cents,  what  will  18 
pounds  cost  at  the  same  rate? 

Ibs.     Ibs.         cts. 
Thus,     3   :  18   :   :  25 
18 

200 
25 

3)450 

$1.50  Answer. 


110  SINGLE    RULE   OP    THREE. 

2.  If  7  pounds  of  coffee  cost  b?i  cents,  what  must  I 
pay  for  244  pounds? 

Ibs.     Ihs.          cts. 
Thus,    7   :  244   :   :  87J 


1708 
1952 
122 

7)21350 


$30.50  Answer. 


3.  If  450  barrels  of  flour  cost  $1350,  what  will  8  bar- 
rels cost? 

Ibis.     bbls.         $  $ 

Thus— As  450     :    8   :   :  1350   :  24,  Answer  * 

4.  If  15  yards  of  cloth  cost  £5,  what  number  of  yards 
may  be  bought  for  £125? 

£       £  yds.     yds. 

As  6    :  125    :     :    15    :  3121  Answer. 

5.  If  twelve  men  can  do  a  piece  of  work  in  20  days, 
in  what  time  will  18  men  do  it? 

m.       m.  d.        d. 

As  18   :  12   :    :  20   :  13|  Answer. 

6.  What  will  be  the  cost  of  17  tons  of  lead,  at  223 
dollars  66  cents  per  ton? 

T.     T.          D.  ctfs.        D.  cts 
As  1    :  17   :   :  223.66   :  3802.22  Ans. 

7.  What  will  72  yards  of  cloth  cost  at  the  rate  of  9 
yards  for  £5  12s. 

yds.     yds.         £  s.        £  s. 
As  9    :  72     :   :  5  12   :  44  16  Ans. 

8.  If  750  men  require  22500  rations  of  bread  for  a 
month,  what  will  a  garrison  of  12CO  require? 

Ans.  360CO. 


I  *The  sum  in  the  third  example  is  read  thus: — As  450  is  to  8 
so  is  1350  to  the  answer.  This  is  the  manner  of  reading  all  sunui 
when  stated  in  the  Rule  of  Three. 


SINGLE   RULE   OF    THREE.  Ill 

9.  What  must  be  the  length  of  a  board  that  is  9  inches 
in  width,  to  make  a  surface  of  144  inches,  or  a  square 
foot?  Ans.  16  inches. 

10.  How  many  yards  of  matting  2  feet  6  inches  broad, 
will  cover  a  floor  that  is  27  feet  long,  and  20  broad? 

Ans.  72  yards. 
I   11.  If  a  person's  annual  income  be  520  dollars,  what 
iis  that  per  week?  Ans.  10  dollars. 

12.  If  a  pasture  be  a  sufficient  for  3000  horses  for  18 
days,  how  long  will  it  be  sufficient  for  2000? 

H.          H.  D.      D. 

As  2000   :  3000   :    :    18   :   27  Ans. 

13.  What  must  be  the  length  of  a  piece  of  land    13J- 
rods  in  breadth,  to  contain  one  acre? 

Ans.  11  rods,  4yds,  2ft.  0-^in. 

14.  If  8  men  can  build  a  tower  in  12  days,  in  what 
time  can  12  build  it? 

M.      M.          D.      D. 
As  12     :    8    :     .    12   :   8  Answer. 

15.  If  a  piece  of  land  be  5  rods  in  width,  what  must 
I  be  its  breadth  to  make  an  acre? 

R.      R.          R.      R. 
As  5     :  160   :   :    1     :    32  Answer. 

16.  How  much  carpeting  that  is  lj  yards  in  breadth, 
will  cover  a  floor  7-J-  yards  in  length,  and  five  yards  in 
breadth? 

By  Decimal  Practlons. 
yds.     yds.  yds. 

As  1.5    :    5     :   :    7.5   :    25  Answer.. 

17.  What  will  one  quart  of  wine  cost  at  the  rate  of  12 
dollars  for  16  gallons? 

gals.       qts.       qt.  D.         cts. 

As    16  or  64    :     1    :   :  12.00   :   18f  Ans. 

18.  If  10  pieces  of  cloth,  each  piece  containing  42 
yards,  cost  531  dollars  30  cents,  what  does  it  cost  per 
yard?  Ans.  $1.2i3i 

19.  If  a  hogshead  of  brandy  cost  78  dollars"?  5  csnts, 
what  must  be  given  for  5  gallons  at  the  same  rate? 

Ans.  $6.25 


112  SINGLE    RULE    OF    THREE. 

20.  If  a  staff  4  feet  in  length,  cast  a  shade  on  level 
ground,  8  feet  in  length,  what  is  the  height  of  a  tower, 
whose  shade,  at  the  same  time,  measures  200  feet? 

•ft-      ft-          ft-      ft- 

As  8   :  ^00   :   :  4     :   100  Answer. 

21.  I  lent  my  friend  350  dollars  for  5  months,  he  pro- 
mising to  do  me  the  same  favour;  hut  when  requested, 
he  could  spare  only  125  dollars.     How  long  ought  I  to 
keep  it  to  balance  the  favour? 

D.        D.  M.      M. 

As  125   :  350   :     :    5     :    14  Answer. 

22.  If  7  oxen  he  worth  10  cows,  how  many  cows  will 
21  oxen  be  worth? 

Ox.      Ox.          C.      C. 
As  7     :    21     :     :   10   :  30  Answer. 

23.  If  board  for  1  year,  or  52  weeks,  amount  to  $182, 
what  will  39  weeks  come  to?  Ans.  $133.50. 

24.  If  30  bushels  of  rye  may  be  bought  f>r  120  bushels 
of  potatos,  how  many   bushels  of  rye  imy  be  bought  for 
300  bushels  of  potatos?  An«.  150  bushels  rye. 

25.  A  gentlemin  bought  a  bag  of  coffee  for  the  use  of 
his  family,  weighing  127  Ibs.,  for  which  he  gave  $15.- 
25  cts.;    what  was  it  a  pound?  Ans.  12-}- cts. 

26.  If  a  gentleman  spends   4  dols.  62i  cts.  every  day 
how  much  will  that  amount  to  in  a  year? 

Ans.  1688.124, 

27.  A  lady  bought  for  the  use  of  her  family,  a  piece 
of  cloth,  containing  16  yds.  3  qrs.  2  na.,  at  $1.25  cts 
a  yard;  what  was  the  amount  of  the  whole? 

Ans.  $21.09+ 

28.  A,  failing  in  trade,  owes  the  following  sum=,  viz 
to  B  $1600.60,  to  C  $500,  to  D  $750.20,  to  E  $1000 
to  F  $230;  and  his  property,  which  is  worth  no  more 
than  $1020.20,  he  gives  up  to  his  creditors;  how  mucl 
does  he  pay  on  the  dollar?   arid  what  is  the  amount  oi 
loss  sustained  by  all? 

Ans.  25  cts.  on  $1 ;  and  the  amount  of  loss  is  $3060.60 

29.  A  farmer   made  from  an  orchard  of  apples,  146 
barrels  of  cider,  which  he  afterwards  sold  at  $3.12j  cts 
a  barrel;  what  was  the  amount  of  the  whole? 

Ans.  $456.25. 


SINGLE   RULE   OF    THREE.  113 

30.  A  flour  merchant  sold  1574  bbls.  of  flour,  at  $5.12i 
cents  a  barrel,  what  was  the  amount?     Ans.  $8066.75. 

31.  A  merchant  failing,  was  able  to  pay  his  creditors 
only  62 J-  cents  in  a  dollar;  how  much  wi.l  a  person  re- 
ceive, whose  claim  is  746.25  cents? 

Ans.  $466.40,6,25. 

32.  A  lady  purchased  a  set  of  silver,  weighing  in  the 
whole  5  Ib.  6  oz.  5  dwt.  at  1.50  cents  an  ounce;  what 
was  the  cost  of  the  whole?  Ans.  $93.37,5. 

33.  A  lady,  intending  to  make  a  bed  quilt,  containing 
8-J-  square  yards,  desired  her  daughter  to  inform  her  how 
much  muslin  l£  yard  wide  would  be  required  to  line  the 
same;  how  many  did  it  take?     Ans.  6.8  yds.  or  6-}  yds. 

34.  A  certain  field  will  afford  pasture  for  10  oxen,  for 
60  days;  how  long  will  the  same  pasture  suffice  for  24 
oxen?  Ans.  25  days. 

35.  A  pipe  will  drain  off  a  cistern  of  water  in  12  hours; 
how  many  pipes  of  the  same  size  will  empty  it  in  30 
minutes?  Ans.  24  pipes. 

36.  Two  persons  travel  on  the  same  road,  and  in  the 
same  direction.     One  sets  out  5  days  before  the  other, 
and  travels  20  miles  a  day;  the  other  travels  25  miles 
a  day;   how  long  before  the  latter  will  overtake  the  for- 
mer? Ans.  20  days. 

37.  If  48  men  can  build  a  fortification  in  24  days,  how 
many  men  can  do  the  same  in  192  days?  Ans.  6  men. 

38.  A  certain  piece  of  work  was  done  by  120  men  in 
8  months,   how  many  men  will  it  take  to  do  another 
piece  of  work  of  the  same  magnitude  in  2  months? 

Ans.  460. 

39.  A  merchant  failing  in  trade,  owes  29475  dollars: 
he  delivers  up  his  property,  which  is  worth  21894  dol 
lars  3  cents;  how  much  does  this  sum  pay  on  the  dollar 
towards  what  he  owes?  Ans,  74cts.  2m.-{- 

40.  If  property  rated  at  28  dollars,   pay  a  tax  of  21 
dollars,  how  much  is  that  on  the  dollar? 

Ans.  75  cents. 

41.  If  i  of  a  yard  cost  62i  cents,  what  will  f  of  a  yard 
cost?  Ans.  $2.187.+ 


114  SINGLE  RULE  OF  THfiEE 

42.  If  a  tax  of  30,000  dollars  be  laid  on  a  town  in 
which  the  ratable  property  is  estimated  at   9,000,000 
dollars,  what  will  be  the  tax  of  one  of  the  citizens  whose 
ratable  estate  is  reckoned  at  750  dollars? 

D.  D.  D.        D.  cts. 

As  9,000,000   :  30,000   .  :   750   :     2   :  50 .Ans* 

43.  How  far  are  the  inhabitants  on  the  equator  carried 
n  a  minute,  allowing  the  earth  to  make  one  revolution 
.n  24  hours;  and  allowing  a  degree  to  contain   6£J 
miles? 

The  earth  being  divided  into  360  degrees,  allowing 
69*  miles  to  a  degree,  makes  the  distance  round  it  to  be 
25020  miles; — the  number  of  minutes  in  24  hours  is 
1440.  Ans.  17  mile?,  3  fur. 

44.  There  is  a  cistern  having  4  spouts;  the  first  will 
empty  in  15  minutes,  the  second  in  thirty  minutes,  the 
third   in  45  minute?,  and  the  fourth  in  60  minutes:  in 
what  time  would  the  cistern  be  emptied,  if  they  were 
all  running  together? 

As  15   :  1    :  :  CO   :  6 

30   :   1    :  :  £0   :  3 

45   :  1    :  :  90   :  2 

60   :  1    :  :  SO   :  1-} 


cisterns,  cist.       mln.  rnin.  sec. 
Then,  decimally,  as  12.5  :  1    :   :  90   :     7  . 12   Ans 

*In  making  taxes  in  a  due  proportion,  according  to  the  value 
of  each  man's  ratable  estate,  proceed  in  the  following  manner. 
Make  the  amount  of  ratable  property  the  first  term ;  make  the 
sum  to  be  raised  the  second  term ;  and  one  dollar  the  third  term 
and  the  number  arising  from  this  operation  will  be  the  amount  to 
be  raised  on  the  dollar,  From  thi-,  make  a  tax  table  from  one 
dollar  to  30,  or  any  amount  necessary.  In  the  same  manner  fine 
what  is  to  be  paid  on  a  cent  of  ratable  estate ;  and  from  this 
make  a  table  from  1  to  99  cents;  then,  from  these  tables,  take 
each  man's  tax.  Thus,  if  the  tax  were  75  cents  on  the  dollar 
and  you  would  know  what  a  portion  of  property  pays,  that  i 
rated  at  $28,80,  the  tables  will  show  the  amount  to  be  $21,  fo 
the  dollar?,  and  60  cts.,  for  the  cents.  In  estimating  property 
for  making  taxes,  it  is  customary  to  rate  it  much  lower  than  its 
real  value. 


DOUBLE    RULE    OF    THREE. 

45.  If  a  ship^s  company  of  15  persons  have  a  quantity 
of  bread,  sufficient  t-j  afford  to  each  one  8  ounces  per 
day,  during  a  voyage  at  sea,  what  ought  to  be  their  al- 
lowance, under  the  same  circumstances,  if  5  persons  be 
added  to  their  number?  Ans.  6 ounces. 

Note. — As  the  Rule  of  Three  in  Vulgar  and  Decimal 
Fractions  require  the  same  statements  as  in  whole  num- 
bers, and  is  performed  by  multiplication  and  division 
after  the  same  manner  of  other  sums  in  the  Rule  of 
Three,  it  is  deemed  unnecessary  to  give  any  examples. 
When  the  pupil  understands  Fractions  and  the  Rule  of 
Three,  he  will  find  no  difficulty  with  the  Rule  of  Three 
in  Fractions. 
Q.  1.  What  is  the  Rule  of  Three  sometimes  called? 

2.  What  does  it  teach? 

3.  Which  of  the  terms  must  be  set  in  the  third  place? 

4.  How  do  you  ascertain  which  ought  to  be  the  first 

term,  and  which  is  the  second? 

5.  If  the  third  term  consist  of  different  denomina- 

tions, what  do  you  do  with  them? 

6.  What  do  you  do  if  the  first  and  second  terms  are 

of  different  denominations? 

7.  After  stating  the  sum,  and  reducing,  when  neces- 

sary, the  terms  to  similar  denominations,  how 
do  you  proceed  to  do  the  sum? 
8.  How  are  sums  in  the  Single  Rule  of  Three  proved? 


DOUBLE  RULE  OF  THREE. 

The  Double  Rule  of  Three  is  that  in  which  five  or 
more  terms  are  given  to  find  another  term  sought. 

RULE. 

Set  the  term  which  is  of  the  same  denomination  aa 
:he  term  sought,  in  the  third  place;  then  consider  each 
pair  of  similar  terms  separately,  and  this  third  one,  as 
making  the  terms  of  a  statement  in  the  Single  Rule  of 
Three,  setting  the  similar  terms  in  the  first  or  second 
places,  according  to  the  rule  of  the  Single  Rule  of  Three. 
After  stating  the  question  in  this  manner,  and  reducing, 


116 


DOUBLE    RULE    OP    THREE. 


if  necessary,  the  similar  terms  to  similar  denominations, 
then  multiply  the  terms  in  the  second  and  third  places 
|together  for  a  dividend,  and  the  terms  in  the  first  place 
together  fora  divisor — the  quotient,  after  dividing,  will 
be  the  term  sought. 

Sums  in  this  rule  may  also  he  done  by  two  or  more 
statements  in  the  Single  Rule  of  Three. 
PROOF. 

By  inverting  the  statement,  or,  more  easily,  by  two 
statements  in  the  {Single  Rule  of  Three. 

EXAMPLES. 

1.  If  8  men,  in  16  days,  can  earn  96  dollars,  how 
much  can  12  men  earn  in  26  days? 
men      8     :     12     :     : 
days    16     :     26     :     : 

128        312 


1872 
2808 

D. 

128)29952(234  Answer. 
256 

435 
384 


512 
512 

2.  If  $100  gain   $8  in  12  months,  what  will  $400 
gain  in  9  months? 

As  100  :  6   :    •  J 

months     12  :  9  :   :   \ 

1200    54 

400  X 

12|00)216|00 


$18  Answer. 


DOUBLE    RULE    OF    THREE.  117 

3.  If  16  men  can  dig  a  trench  54  yards  in  length  in  6 
days,  how  many  men  will  be  necessary  to  complete  one 
135  yards  in  length,  in  8  days? 

By  two  statements  in  the  Single  Rule  of  Three. 

yds.       yds.  men.  men. 

As  54     :   135      :   :     16   :  40. 
days.  ds.         men.  men. 
Then,  as  8:6::     40  :  30  Answer. 

4.  If  $100  in  one  year  gain  $5  interest,  what  will  be 
the  interest  of  $750  for  7  years?          Ans.  $262.50. 

5.  If  9  persons  expend  $120  in  8  months,  how  much 
will  24  oersons  spend  in  16  months  at  the  same  rate? 

Ans.  $340. 

6.  If  54  dollars  be  the  wages  of  8  men  for  14  days, 
what  mast  be  the  wages  of  28  men  for  20  days  at  the 
same  rate?  Ans.  $270. 

7.  If  a  horse  travel  130  miles  in  3  days,  when  the 
;lays  are   12  hours  in   length,  in  how  many  days  of  10 
hours  each  can  he  travel  360  mile^s?     Ans.  9$ f  days. 

8.  If  60  bushels  of  corn  can  serve  7  horses  28  days, 
how  many  days  will  47  bushels  serve  6  horses? 

Ans.  25f  £  days. 

9.  If  a  barrel  of  beer  serve  7  persons  for  12  days, 
how  many  barrels  will  be  sufficient  for  14  persons  for  a 
year,  or  365  days?  Ans.  6C|  barrels. 

10.  If  8  men  spend  32  dollars  in  13  weeks,  what  will 
24  men  spend  in  52  weeks?  Ans.    $384. 

|Q.  1.  How  rmny  terms  are  generally  given  in  the  Dou- 
ble Rule  of  Three? 

2.  Which  of  the  terms  must  be  set  in  the  third  place? 

3.  How  do  you  ascertain  which  of  the  other  terms 

should  he  placed  in  the  first,  and  which  in  the 
second  place? 

4.  Which  of  the  terms  do  you  multiply   together  for 

a  dividend? 

5.  How  do  you  form  a  divisor? 

6.  How  do  you  proceed  when  the  terms  consist  of  dif- 

ferent denominations? 

7.  How  is  a  sum  in  the  Double  Rule  of  Three  proved  ? 


i  io  DOUBLE    RULE    OF    '1HKLE. 

Promiscuous  questions  in  Simple  and  Compounl  Pro- 
portion. 

1.  Wh^,t  can  you  buy  15  tons  of  hay  for,  if  3  ton: 
cost  $36?  "Ans.  $150. 

2.  William's  income  is  $1500  a  year,  and  his  daily 
expenses  are  $2.50;  how  much   will  he  have  saved   ;  t 
the  year's  enci?  Ans.  $567.50. 

3    Jf  7  men  can  reap  84  acres  of  wheat  in  12  davs; 
how  many  can  reap  100  acres  in  5  days?  Ans.  20  men. 

4.  If  a  horse  will    trot  in  a  gig  8  miles  in  an   hour, 
how  far  will  he  trot  at  the  same  rate,  in  3i  hours? 

Ans,  28  miles^ 

5.  A  merchant  bought  5  pieces  of  muslin,  each  con- 
taining 26  yards,  at  11  cents  a  yard;   what  did  they 
amount  to?  Ans.  $14.30. 

6.  If  a  family  of  9  persons,  in  24  months,  spend  $480, 
how  much  would  16  persons  spend  in  8  months? 

Ans.  $320. 

7.  A  merchant,  owning  |  of  a  vessel,  sells  T  of  his 
share  for  $500;  what  was  the  whole  vessel  worth? 

|  O/"|=JL=J;  then,  as  2.   of  the  vessel  is  $500,  -i  is 
$250,  and  j,  or  the  whole  vessel,  is  5x*250=$1250. 
Or  thus;  f  o/f   :  1    :   :  5CO   :  $1250. 

Ans.,  as  before. 

8.  If  li  Ib.  indigo  cost  $3.84,  what  will  49,2  ll:s.  cost 
|at  the  sariKx  rate?  Ans.  $125,952. 

9.  If  1 12  acres  of  meadow  be  mowed  over  by  16  men. 
in  7  days;  how  many  acres  can -24  men  mow  over,  in 
19  days?  Ans.  456  acres. 

10.  If  8  cwt.  of  iron  can   be  carried    128  miles  for 
$12.80,  what  will  be  the  expense  of  carrying  4  cwt.  32 
miles?  Ans.  $1.60. 

11.  A  merchant  bought  a  bale  of  cloth,  containing 
375  yds.  at  ^  3.12^  a  yard;  what  did  the  whole  amount 
t©?   "  Ans.  $1171.87,5. 

12.  A  mother  allows  her  daughter,  at  a  hoarding  school, 
3  cents  a  day  for  spending  money;  how  much  will  that 
amount  to  in  a  year?  Ans.  $10.95. 

13.  Suppose  the  wages  of  6  persons  fjr  21  weeks  be 
288  dollars,  what  must  14  persons  receive  for  46  weeks  \ 

Ans.  $1472. 


PRACTICE.  11<J 

14.  If  1  hundred  weight  of  sugar  cost  13  dollars  50 
cents,  what  must  be  paid  for  ITcwt.  3qrs.  141b.? 

Ans.   $'241.31-}  cents. 

15.  How  many  yards  of  paper,  2i  feet  wide,  will  be 
required  to  cover  a  wall,  which  is  12  feet  long,  and  9| 
fee   high?  Ans.   14yds.  1ft.  Sin. 

1  .  If  it  take  5  men  to  make  150  pair  of  shoes  in  20 
days>  how  many  men  can  make  1350  pair  in  60  days? 

Ans.  15  men. 

17.  If  a  footman  travel  240  miles  in  12  days,  when  the 
days  are  12  hours  long;  how  many  days  will  he  require 
to  travel  720  miles,  when  the  days  arc  16  hours  long? 

Ans.  27  days. 

18.  If  4  men  receive  $24  for  6  days1  work,  how  much 
will  8  men  receive  for  12 days'  work?         Ans.  $96. 

19.  If  9  persons  in  a  family  spend  $1512  in  1  year  (or 
12  mo.),  how  much  will  3  of  the  samo  persons  spend  in 
4  months?  Ans.  $168. 

20.  A  regiment  of  soldiers,  consisting  of  800  men,  arc 
to  be  clothed,  each  suit  containing  4|-  yds.  of  cloth, 
which  is  If  yd.  wide,  and  lined  with  flannel  J  yd.  wide; 
how  many  yards  of  flannel  will  be  sufficient  to  line  all 
the  suits?  Ans.  8633  yds.  1  qr.  lj  na. 


PRACTICE. 

Practice  is  a  short  method  of  doing  all  sums  in  the 
Single  Rule  of  Thiee,  that  have  one  for  their  first  term, 
and  is  of  great  use  among  merchants. 

It  may  be  proved  by  Compound  Multiplication,  or  by 
the  Single  Rule  of  Three. 

Questions  to  prepare  the  learner  for  this  rule. 

1.  What  will  50  yards  of  tape  cost  at  i  of  a  cent  per 

yard? 

2.  What  will  40  pounds  of  beef  come  to  at  1  of  a  cent 

per  pound? 

3.  What  will  100  figs  come  to  at  J  of  a  cent  a  piece? 

4.  How  many  pence  will  40  peaches  come  to  at  one 

farthing  a  piece? 

5.  How  many  shillings  and  pence  will   52  peaches 

come  to  at  one  farthing  a  piece? 


120 

PRACTICE. 

PRACTICE  TABLE,   OR   TABLE  OF  ALIQUOT  PARTS. 

ds. 

dots. 

*.  A      £• 

d.         s. 

50 

=3 

.1.    •* 

i 

10  0  z=  i  ' 

} 

1                 1    "\     O 

25 

i 

c 

68        | 

H~  "  i  7 

20 

i 

9 

50        j 

2. 

2    *  i  s- 

12*       \ 

>    2~ 

4  0 

P 

3         jf*! 

6 

1 

A 

jr 

3  4 

"5 

4         |   j  jf- 

5 

-j 

2  f>         J 

6          v  J   ' 

4 

1  8        TV 

5 

^r5.     Ib.     cwt. 

771. 

c~ts. 

i  o      A  . 

a  or  56  •=  i  ^ 

5 

2 
1 

*1  s> 

.A)? 

gr.            d. 

d. 

1         28       I 
16       | 
14      J 

0 

8     TV 

1 

CASE 

I. 

rV 

I  When  the  price  of  one  yard,  pound,  $c.  is  in  farthings. 

BULK 

Divide  by  the 

aliquot  parts 

of 

a 

penny,  and  the 

an- 

swer  will  be  in  pence,  which  reduce 

to  shillings,  pounds, 

EXAMPLES. 

1.  What 

is  the  value  of 

2. 

What  is  the  value  of 

380  at  one  farthing  each? 

744  at  3  farthings  each? 

1 

J 

380 

2 

i 

744 

J 

12)  )5 

1 

i 

372 

- 

— 

186 

•v 

7s. 

lid.  Ans. 

f 

12)358 

3.  What  is  the  value  of 

460  at 

2  farthings  each? 

20)46—  G 

2 

i 

460 

— 

£2  6.  6d.  Ans. 

12)230  pence. 

19s.  2d.  Answer. 

4.  What 

is  the  worth  of  298  at  Jr 

!.?        An?.  6s.  2id. 

5.  What 

is  the  worth  of  586  at  id 

.?    Ans.  £1.  4s. 

5d. 

|     6.  What 

is  the  worth  of  964  at 

fd.?    Ans.  £3.  Os, 

3d. 

PRACTICE. 


121 


CASE  II. 

When  ike  price  is  any  number  of  pence  less  than  12. 

RULE. 

Divide  by  the  aliquot  parts  of  a  shilling,  and  the  an- 
swer will  be  in  shillings,  which  may  be  reduced  to 
pounds. 

EXAMPLES. 
I.  II. 


672  at  Id. 
20)56 
£2.  16s.  Ans. 


444at2d. 
20)74 


3.  What  is  the  value  of  237  at  3d.? 

4.  What  is  the  value  of  594  at  4d.? 

5.  What  is  the  value  of  868  at  6d.? 

6.  What  is  the  value  of  988  at  5d.? 

7.  What  is  the  value  of  1049  at  8d.? 


£3  14s.  Ans. 

£  s.  d. 
2  19  3 
9  18  0 
14  0 
Ans.  20  11  8 
Ans.  34  19  4 


Ans. 
Ans. 
Ans.  21 


8.  What  is  the  value  of  1294  at  10d.?  Ans.  53  18  4 

CASE  III. 

When  the  price  in  pence  exceeds  the  number  of  12. 
RULE. 

Consider  the  number  given  in  the  sum  as  containing 
so  many  shillings.  Then  divide  by  such  aliquot  parts 
as  may  be  formed  by  the  pence  over  a  shilling,  adding 
the  product  to  the  sum.  The  answer  will  be  in  shillings. 
EXAMPLES. 


20 


600 
75 

675 


at  13jd. 


£33  15s.  Answer. 

Note. — In  this  example,  I  consider  the  sum  as  600 
shillings.  Then,  as  the  given  price  is  1  Jd.  over  a  shil- 
ling, which  makes  J  of  a  shilling,  I  divide  the  sum  by  8, 
and  add  the  quotient  to  the  given  sum  j  which  makes  675 
shillings,  or  £33  15s. 


122 


PRACTICE. 


2.  What  is  the  worth  of  450  at  14d.?    Ans.  £26.  5s. 

3.  What  is  the  worth  of  570  at  16d.?   Ans.  £38.  Os. 

CASE  IV. 

When  the  price  is  any  number  of  shillings  under  20. 

RULE. 

Divide  by  the  aliquot  parts  of  a  pound,  and  the  answer 
will  be  in  pounds.  Or,  consider  the  sum  as  being  so 
many  shillings,  then  multiply  the  sum  by  the  number 
of  shillings  in  the  price.  The  product  will  be  the  an- 
swer in  shillings;  which  reduce  to  pounds. 
I.  n. 


5s. 


1296  at  5s. 


£324  Ans. 


723     at  12s 
12  X 


20)3676 


£433.  16s.  Answer. 
The  second  example  is  done  by  the  second  method, 
which  is  thought  by  many  to  be  the  easier  way. 

£ 

3.  What  is  the  value  of  1128  at  3s.?  Ans.     169.  4 

4.  What  is  the  value  of   889  at  4s.?  Ans.     177.  16 

5.  What  is  the  value  of  1616  at  9s.?  Ans.     727.  4 

6.  What  is  the  value  of  2868  at  18s.?  Ans.  2581. 

CASE  V. 
When  the  price  is  in  pounds,  shillings  and  pence. 

RULE. 

Multiply  the  sum  or  quantity  by  the  number  of  pounds 
in  the  price,  then  divide  the  aliquot  parts  of  shillings 
and  pence,  and  add  the  quotients  to  the  product — theii; 
sum  will  be  the  answer. 

I.  EXAMPLES.  II. 


10 


448at£410s.6d.4 
4 


£2027.  4s.  Ans. 


5678  at  £7  4s.  9d 

7 


39746 
1135  12 
141  19 
70  19  6 


£U094.  10s.  6d. 


Note. — In  the  second  example,  after  multiplying  the 
sum  by  the  number  of  pounds,  as  4s.  is  i  of  a  pound,! 
divide  by  5,  which  gives  1135  pounds  in  the  quotient; 
and  leaving  a  remainder  of  3  pounds,  which  reduced  to 
shillings  and  divided  by  5  give  12s.  Then,  as  6  and  3 
make  9,  the  number  of  pence,  and  as  6d.  is  J  of  4s.,  I 
divide  the  quotient  by  8,  which  gives  141  pounds  wish 
a  remainder  of  7;  this  being  reduced  to  shillings,  and 
the  12  shillings  above  added  to  it  make  152,  which  di- 
vided still  by  the  8  give  19  shillings.  And  as  3  is  1  of 
6,  or  its  aliquot  part,  I  divide  the  hist  quotient  bv  2  — 
This  gives  70  pounds  and  a  remainder  of  I,  which  is 
20  shillings;  and  adding  it  wilh  19  shillings  above,  the 
amount  is  39  shillings.  This  divided  by  the  2  gives  11 
shillings  and  a  remainder  of  1  shilling,  or  12  pence; 
which  divided  still  by  the  2,  makes  6d.  And  thus  the 
answer  is  obtained. 

3.  What  is  the  amount  of  288  at  £5.  10?.  4d.? 

Ans.  £1588.  16s. 

4.  What  is  the  amount  of  642  at  £:>.  4s.  6d.? 

Ans.  £51 22.  Cs. 
T>.  What  is  the  amount  of  734  at  £12.  *s.  £d.? 

Ans.  £8905.  17s.  4d. 
CASE  VI. 

When  the  quantity  consists  of  different  denominations,  and 
the  price  is  in  pounds,  shillings,  6$c. 

RULE. 

Multiply  the  price  of  the  highest  denomination  given, 
by  the  whole  of  the  highest  denomination,  then  divide  by 
aliquot  parts  of  each  of  the  lower  denominations  in  the 
sum.  Add  the  results  together,  and  their  sum  will  be 
the  answer.  EXAMPLES. 

i.  £    s.    d. 


3  cwt.  2  qrs.  14  Ibs. 
2  qrs.  are  ^  of  a  cwt.. 


14  Ihs.  are  1  of  2  qrs. 


at  4     6     2  per  cwt 
3 


12  18     6 
2     3     1 
10     9i 

£15.   12s.  4id.  Ans. 


124  FELLOWSHIP. 

3.  4  cwt.  3  qrs.  12  Ibs.  at  £8.  4s.  4d.  per  cwt. 

Ans.  £39.  18s.  2d. 

3.  5  cwt.  3  qrs.  4  Ibs.  at  £9.  6s.  8d.  per  cwt. 

Ans.  £54.  0.  0. 

4.  7  cwt.  0  qr.  14  Ibs.  at  £2.  3s.  4d.  per  cwt. 

Ans.  £15.  8s.  9d. 

5.  8  cwt.  3  qrs>  24  Ibs.  at  £1.  2s.  3d.  per  cwt. 

Ans.  £9.  19s.  5id. 

6.  9  cwt.  1  qr,  18  Ibs.  at  £3.  10s.  lOd.  per  cwt. 

Ans.  £33.  6s.  7d. 

7.  10  cwt.  2  qrs.  10  Ibs.  at  £4.  4s.  6d.  per  cwt. 

Ans.  £44.  14s,  &}d. 
1 1.  What  is  practice? 
2*  Wherein  is  it  particularly  useful? 
3*  Repeat  the  table  of  aliquot  parts. 

4.  How  many  cases  are  there  in  poundsj  shillings, 

&.C.! 

5.  Repeat  the  rule  of  each  different  case. 

6.  How  are  sums  in  practice  proved? 


FELLOWSHIP. 

Fellowship  is  an  easy  rule  by  which  merchants  or 
other  persons  in  company,  are  enabled  to  make  a  just 
division  of  the  gain  or  loss  in  proportion  to  each  person's 
share.  Sums  in  Fellowship  are  generally  done  by  the 
Rule  of  Three. 

CASE  I. 

When  the  several  shares  are  considered  without  regard 

to  time. 

RULE. 

As  the  sum  of  all  the  stock  is  to  each  person's  partic- 
ular share  of  the  stock^  so  is  the  sum  of  all  the  gain  or 
loss,  to  the  gain  or  loss  of  each  person, 

PROOF, 

Add  together  all  the  shares  of  gain  or  loss,  and  if  it  be 
right,  the  sum  will  be  equal  to  the  whole  gain  or  loss. 


FELLOWSHIP,  125 

EXAMPLES. 

1.  A  and  B  purchase  certain   gocc's   amounting  to 
|f  580,  of  which  A  pays  $350  and  B  $230.     They  gain 

" — what  is  each  man's  share  of  the  gain? 
A  $350 
B  $230 

A's  share,     gain.       $     cts. 

580     :  350   :    :  262   :  158.10  JJ-  A's  gain. 

B's  share,     gain.       $     cts. 
Then,  as  580    :  230   :    :  262   :  103.8?) jf  B's  gain. 

2.  A,  B  and  C  formed  a  company.     A  put  in  $-10,B 
60  and  C  80.  They  gained  $72: — what  was  each  man's 
share?  Ans.  A  gained  $16,  B  24  and  C  32. 

3.  A,   B  and  C  lose  a  quantity  of  property   worth 
$2400;  of  which  A  owned  J,  B  -J-,  and  the  remainder  to 
C;  what  does  each  lose? 

Ans.  A  loses  $600,  B  800  and  C  1000. 

4.  Three  persons  entered  into  partnership  in  trade 
iThe  first  put  in  250  dollars;  the  second  put  in  350  dol 
liars;  and  the  third  put  in  500  dollars;  and  in  12  months, 
they  found,  by  examining  their  hooks,  that  they  had  gain- 
ed 460  dollars;  how  must  the  gain  be  divided  between 
them,  so  that  each  may  have  his  due  proportion? 

fA's  share,  $104.54,545+ 
Answer.      \  B's  share,      146.3G,363-f 
IC's  share,      209.09,09  + 

5.  A  and  B  purchase  goods  worth  $80,  of  which  A 
pays  30  and  B  50.     They  gain  $20; — what  is  the  gain 
of  each?  Ans.  A's  gain  is  $7.50  and  B's  12.50. 

6.  Four  men  formed  a  capital  of  $3200.     They  gain 
ed  in  a  certain  time  $6560.     A's  stock  was  $560,  B's 
1040,  C's  1200  and  D's  400.     What  did  each  gain? 

Ans.  A  gained  $1148,  B  2132,  C  2460  and  D  820. 

CASE  II. 

When  the  different  stocks  in  company  are  considered  in 
\  relation  to  time. 

RULE. 

Multiply  each  man's  stock  by  the  time  it  has  been  a 
part  of  the  whole  stock;  then,  as  the  sum  of  the  pro- 
lucts  is  to  either  single  product,  so  is  the  whole  sum  of 
or  loss  to  the  gain  or  loss  of  each  man. 


12(1  TAKE    AND    THET. 

EXAMPLES. 

1.  A,  Band'C  hold  a  pasture  in  common,  for  which 
they  pay  $40  per  annum.     A  put  in  9  cows  for  five 
weeks;  B,  12  cows  for  7  weeks;  and  C,  8  cows  for  16 
weeks. — What  must  each  man  pay  for  the  rent? 

9x  5=  45 
12x  7=  84 
8x16=128 

As  257  -  45  :  :  40  :  7^T  A's  part. 
As  257  ;  84  :  :  40  :  13  A,  B's  part. 
As  257  :  128  :  :  40  :  19f  f  ?  C's  part. 
2.  A  with  a  capital  of  £1000.,  entered  into  business 
on  the  first  of  January.  On  the  first  of  March  follow- 
ng  he  took  in  B  as  a  partner,  who  brought  with  him  a 
capital  of  £1500.;  and  three  months  after  they  are 
joined  by  C,  with  a  capital  of  £2800.  At  the  end  of 
the  year  they  find  they  have  gained  £1776.  10s.  How 
must  it  be  divided  among  them? 

Ans.  A^s  part  will  be  £457.     9s.     4£d. 
B's  part  will  be  £571.  16s.     8-jd. 
C's  part  will  be  £747.     3s.  lljd. 
Q.I.  What  is  Fellowship? 

2.  By  what  rule  are  sums  in  Fellowship  usually  done  ? 

3.  How  do  you  proceed  when  the  shares  are  consider- 

ed without  regard  to  time  ? 

4.  How  do  you  proceed  when  the  shares  are  consider- 

ed in  relation  to  time  ? 

5.  How  are  sums  in  Fellowship  proved? 

TARE  AND  TRET. 

Tare  and  Tret  are  certain  allowances  made  by  mer- 
chants in  selling  their  goods  by  weight. 

Tare  is  an  allowance  made  for  the  weight  of  the  bar- 
el,  bag,  &c.,  that  contains  the  article  or  commodity 
bought. 

Tret  is  an  allowance  of  4  Ibs.  in  every  104  Ibs.  for 
vaste,  dust,  &c. 


TARE    AND    TRET.  127 

Gross  weight  is  the  weight  qf  the  goods,  together  with 
the  barrel,  box,  or  whatever  contains  them.  When  the 
tare  is  deducted  from  the  gross,  what  remains  is  called 
suttle. 

Neat  weight  is  the  weight  of  articles  after  all  allow- 
ances are  deducted, 

CASE  I. 

When  iJie  tare  is  so  much  in  the  whole  gross  weight. 
RULE. 

Subtract  the  tare  from  the  quantity — the  remainder 
will  be  the  neat  weight. 

EXAMPLES. 

In  Ghhds.  of  sugar,  each  weighing  9  cwt.  2  qrs.  10  Ibs. 
gross,  tare  25  Ibs.  per  hhd.  how  much  neat  weight? 

cwt.     qr.     Ib.  cwt.     qrs.     Ibs. 

25x6=1        1       10  tare         9      2       10 

6X 


57      2        4  gross, 
1       1       10  tare 


50       0       22Ans. 

2.  What  is  the  neat  weight  of  456  cwt.  1  qr.  19  Ibs. 
of  tobacco,  tare  in  the  whole  15  cwt.  2  qrs.  13  Ibs?  , 
Ans.  440  cwt.  3  qrs.  6  Ibs. 
3  What  is  tho  neat  weight  of  5  casks  of  sugar,  the 
gross  weight  and  tare  as  follows? 

cwt.  qrs.  Ibs.         qrs.  Ib. 
No.   1.  Gross  4     2     14  Tare  1     5 

2.  3     0     17  1     1 

3.  _  5     3     10  2  11 

4.  6     1     16  2  27 

5.  3.2     18  * 1     3 


'    Ans.  21  cwt.  2  qrs. 
CASE  II. 
When  the  tare  is  at  so  much  per  cwt. 

RULE, 

Divide  the  gross  weight  by  the  aliquot  parts  of  a  cwt 
then  subtract  the  quotient  from  the  gross,  and  the  re- 
mainder will  be  the  neat  weight... 


128  TARE    AND    TRET. 

EXAMPLES. 

1.  In  129  cwt,  3  qrs.  16  Ibs.  gross,  tare  14  Ibs.  per 
cwt,  what  neat  weight? 


14  Ibs 


129    3     16    gross. 
16     0     26i 


113     2     171  Answer. 
2.  In  97  cwt.  1  qr.  7  Ibs.  gross,  tare  20  Ibs.  per  cwt. 
what  neat  weight? 

Ans.  79  cwt.  1  qr.  20J  Ibs. 

2.  What  is  the  neat  weight  of  35  kegs  of  raisins,  gross 
weight  37  cwt.  1  qr.  20  lb.; — tare  per  cwt.  14  Ibs.? 

Ans,  32  cwt.  3  qrs. 

3.  What  is  the  neat  weight  of  6  hogsheads  of  sugar, 
each  weighing  8  cwt.  2  qrs.   14  Ibs.  gross;  tare  16  Ibs. 
per  cwt.?  Ans.  44  cwt.  1  qr.  12  Ibs. 

Note. — When  the  tare  per  cwt.  is  not  an  aliquot  part, 
the  tare  may  be  found  by  the  Rule  of  Three,  thus — As 
112  is  to  the  number  of  pounds  gross,  so  is  the  rate  per 
cwt.,  to  the  tare  required. 

4.  What  is  the  neat  weight  of  38  cwt.  0  qr.  4  Ibs.  tare, 
at  11  Ibs.  per  cwt. 

cwt.     qr.     Ibs. 
38      0        4=4260  pounds. 
Ibs.         Ibs.  Ibs. 

Then,  as  112  :  4260   :    :    11    :  4 18^  Answer. 
4260 

r4A 

cwt.    qr.  Ibs. 
=  34      1  •  5  f6T82- Answer. 
CASE  III. 

Wlien  tare  and  tret  are  allowed. 

RULE.    ' 

Find  the  tare  according  to  the  preceding  rules,  sub- 
tract it  from  the  gross,  and  the  remainder  will  be  suttle; 
then  divide  the  suttle  by  26,  and  the  product  will  be  the 
tret,  which  subtract  from  the  suttle — the  remainder 
will  be  the  neat. 

Note. — As  4  pounds  on  the  104  Ibs.  is  the  customary  I 
allowance  for  tret,  we  divide  by  26,  because  4  is  •£%  of  j 
104. 


TARE  AND  TRET.  129 

EXAMPLES. 

1.  In  247  cwt.  2  qrs.  15  Ibs-.  gross,  tare  28  Ibs.  per  cwt. 
.nd  tret  4  Ibs.  for  every  104  Ibs.  how  much  neat? 


28  lbs.=  |  J  cwt. 


41bs.=  |  ^  of  104 


cwt.     qr.    Ib.     ox. 
247    2     15 
61     3     17     12  tare  subtract 


185    2    25      4 
7    0     16      0 


Ans.  17S    29      4  neat. 

2.  In  9  cwt.  1  qr.  10  Ibs.  gross,  tare  28  Ibs.  per  cwt. 
and  tret  4  Ibs.  for  every  104  Ibs.  how  much  neat? 

Ans.  6  cwt.  2  qrs.  26    Ibs- 

3.  A  merchant  purchased  4  hhds.  of  tobacco,  weigh- 
ng  as  follows :— The  first  5  cwt.  1  qr.  12  Ibs.  gross,  tare 
35  Ibs.   per  hhd.;  the  2d.  3  cwt.  0  qr.    19  Ibs.  gross, 
;are  75  Ibs.;  the  3d.  6  cwt.  3  qrs. gross,  tare  49  Ibs.;  the 
4th  4  cwt.  2  qrs.  9  Ibs.  gross,  tare  35  Ibs.  and  allowing 
ret  to  each  at  the  rate  of  4  Ibs.  for  every  104  Ibs.    What 
was  the  neat  weight  of  the  whole? 

Ans.  17  cwt.  0  qr.  19  Ibs.  2  oz. 

Exercises  under  the  foregoing  rules. 

1.  There  are  24  hogsheads  of  tobacco;  each  hogshead 
weighs  6  cwt.  2  qrs  17  Ibs.  gross;—  tare  in  all,  17  cwt. 
3  qrs.  27  Ibs.     How  much  will  the  tobacco  amount  to 
at  £1.  10s.  6d.  per  cwt.  Ans.  £216.  Os.  4|d. 

2.  Bought  5  bags  of  coffee,  each  of  which  weighed 
95  Ibs.  gross;  tare  in  the  whole  lOlbs.     How  much  did 
it  amount  to,  at  25  cents  per  pound?     Ans.  $116.25. 

3.  What  is  the  value  of  10  casks  of  alum;  the  whol 
weighing  33  cwt.  2  qrs.  15  Ibs.  gross;  tare  15  Ibs.  per 
cask;  price, 23s.  4d.  per  cwt.?     Ans.  £37.  J3s.6jd. 

4.  A  farmer  sent  a  load  of  hay  to  market,  which  with 
the  cart,  weighed  29  cwt.  3  qrs.  16  Ibs.;  the  weight  of 
the  cart  was  10 J  cwt. ;  what  did  the  hay  come  to,  at  $" " 
a  ton?  Ans.  $14.357+. 


130  SIMPLE    INTEREST. 

5.  A  merchant  bought  sugar  in  a  hogshead,  both  of 
which  weighed  8  cwt.  15  Ibs.;  the  hogshead  alone  weigh- 
ed 1  cwt.  1  qr.;  what  was  the  cost  of  the  sugar,  at  111 
cents  a  pound?  Ans.  $86.73f. 

Q.  1.  What  do  you  understand  by  Tare  and  Tret' 
2.  What  is  tare? 
3    What  is  tret? 

4.  What  is  gross  weight? 

5.  What  is  neat  weight? 

6.  What  is  called  suttle? 


SIMPLE  INTEREST. 

Interest  is  a  premium  paid  for  the  use  of  money  In 
calculating  interest  on  money,  four  things  are  necessary 
to  be  considered,  viz.  the  principal,  the  time,  rate  per 
cent,  and  amount. 

The  principal  is  the  money  lent  for  which  interest  is 
to  be  received. 

The  rate  percent,  per  annum  (by  the  year)  is  the  in- 
:erest  for  100  dollars  or  100  pounds  for  one  year. 

The  time  is  the  number  of  years,  months,  or  days,  for 
which  interest  is  to  be  calculated. 

The  amount  is  the  sum  of  the  principal  and  interest, 
when  added  together. 

Questions  to  prepare  the  learner  for  this  rule. 

1.  If  you  give  $6  for  the  use  of  $100  for  a  year;  how 
much  must  you  give  for  the  use  of  $50? 

2.  If  you  give  $6  for  the  use  of  $100  for  a  year;   how 
much  must  you  give  for  the  use  of  it  for  six  months? — 
How  much  for  three  months? — -How  much  for  4  months? 
How  much  for  8  months? — How  much  for  9  months? 

3.  If  the  interest  of  $200  be  one  dollar  for  a  month; 
how  much  will  it  be  for  15  days? — How  much  for  10 
days? — How  much  for  20  days? 


SIMPLE    INTEREST.  131 


CASE  I. 


When  the  time  is  one  year,  and  the  rate  per  cent,  is  any 
number  of  dollars,  pounds, 

RULE. 

Multiply  the  principal  by  the  rate  per  cent,  divide 
the  product  by  100,  and  the  quotient  will  be  the  interest 
for  one  year. 

EXAMPLES. 

1.  What  is  the  interest  of  328  dollars  for  one  year  at 
6  per  cent.? 

328  In  this  example,  as  cutting  off 

6  the  two  right  hand  figures  is  the 

same  as  dividing  by  100,  the  di 
Ans.  $19.|68cts.          vision  is  omitted. 

2.  What  is  the  interest  of  $9876  for  one  year  at  6  per 
cent.?  6 


$592|56  cts.  Answer. 

When  the  sum  is  in  pounds,  if  there  be  a  remainder 
after  dividing,  or  after  cutting  off  the  two  right  hand 
figures,  the  remainder,  or  figures  cut  off  must  be  reduced 
to  shillings;  and  if  there  be  still  a  remainder  after  di- 
viding the  shillings,  it  must  be  reduced  to  pence,  &c. 

3:  What  is  the  interest  of  £573.  13s.  9£oV^at-6-per 
cent,  per  annum? 

£573.     13s.     9|d.       Note.— When  the  interest  is 
6        for  more  than  one  year,  mul- 

—        tiply  the  interest  for  one  year 

£34|42      2     9        by  the  number  of  years.     To 

20  obtain  the  amount,  the  interest 

must  be  added  to  the  princi- 

8|42  pal. 

12 

5|13  Ans.  £34.  8s.  5d. 

4.  What  is  the  interest  of  £40.  19s.  lid.  3  qrs.  for 
one  year,  at  6  per  cent,  per  annum? 

Ans.  £2.  9s.  2d.  Iqr. 


132  SIMPLE   INTEREST. 

5.  What  is  the  interest  of  87  dollars  for  one  year,  at 
[>  percent,  per  annum?  Ans.  $5.22. 

6.  What  is  the  interest  of  143  dollars  for  one  year,  at 
7  per  cent,  per  annum?  Ans.  $10.01. 

When  the  rate  per  cent,  consists  of  a  whole  number 
and  a  fraction,  as  6j,  6J,  or  6 j,  multiply  the  principal 
>y  the  whole  number,  to  the  product  add  J,  or  |,  as  the 
case  may  be,  of  the  principal  and  then  divide  by  100, 
or  cut  off  the  two  right  hand  figures  as  before. 

7.  What  is  the  interest  of  228  dollars  for  one  year,  at 
per  cent  per  annum? 

$228 


$14|25cts    Answer. 

When  the  principal  consists  of  dollars  and  sents,  mul 
tiply  by  the  rate  per  cent,  without  any  reference  to  the 
separating  point;  then  from  the  product  cut  off  the  first 
right  hand  figure  as  a  fraction  or  remainder,  the  next 
ftgure  will  be  mills,  the  two  next  cents,  and  the  other 
figures,  that  is,  those  on  the  left  of  the  cents,  will  be 
dollars. 

8.  What  is  the  interest  of  $98.79  for  one  year,  at  6 
per  cent,  per  annum?  6 

5|92|7|4  fraction 

Ans.  $5,92c.7m. 

9.  What  is  the  interest  of  432  dollars  73  cents  for  4 
years,  at  6  per  cent,  per  annum? 

$432.73 

6  rate  per  cent. 

259638 

4  number  of  years. 

103|85|5|2  frac.    Ans.  $103.85c.5m 
10.  What  is  the  interest  of  $8420-82  for  three  years 
at  8  per  cent,  per  annum?          Ans.  $2020  99c  6m. 


SIMPLE   INTEREST. 


133 


11.  What  is  the  interest  and  amount  of  $7462.13£  for 
bur  years,  at  7  per  cent  per  annum? 
Ans.  Interest,  $2089.39c.  7m.  AmH.  $9551.53c.2m. 

CASE  II. 

Tojind  the  interest  when  the  given  time  is  months  or  days. 

RULE. 

Find  the  interest  for  one  year,  then  say — as  one  year 
s  to  the  given  time,  so  is  the  interest  of  the  sum  for  one 
rear,  to  the  interest  for  the  time  required.  Or,  instead 
f  the  Rule  of  Three,  it  may  be  done  by  Practice,  thus: 
?or  the  number  of  months,  take  aliquot  parts  of  a  year; 
ind  for  days,  the  aliquot  parts  of  30.* 

EXAMPLES. 

1.  What  is  the  interest  of  $98.50  for  9  months  and  18 
jays,  at  6  per  cent,  per  annum? 
$98.50 
6 


$5.91 100  for  one  year. 
year.    mo.  days.  $  cts.     $  cts.  m. 

Then,as  1     :   9     18     :     :     5    91    :  4  72  8  Ans. 

tn  this  sum,  the  year  is  reduced  to  360  days,  the  0 
months  and  18  days  to  288  days,  and  the  third  term 
stands  as  591  cents. 

The  same  is  done  by  Practice,  thus — 

$98.50 
6 


mo. 
6,  £ofa  year. 

3,  J  of  6  mo. 
15d.£of3mo. 
3,Jofl5ds. 


5.91.0|0 

2.95.5 
1.47.7^ 
24.6J 


Ans.  $4.72.8 


*ln  these  calculations,  a  year  is  reckoned  at  360  days,  and 
month  at  30  days. 


134  SIMPLE    IXTEHEST. 

2.  What  is  the  interest  of  $120.60  for  one  year  and 
ihree  months,  at  6  per  cent,  per  annum? 

Ans.  $9.04c.  5m. 

3.  What  is  the  interest  on  $187.06j  for  10  months, 
at  6  per  cent  per  annum?  Ans.  $9.35c  3rn. 

4.  What  is  the  interest  and  amount  of  640  dollars  for 
4  years  and  7  months,  at  5  pet  cent,  per  annum? 

Ans.  $146.66f  interest.     Arn't.  $786.66j. 

5.  What  is  the  interest  of  $300  for  4  years,  4  months, 
and  20  days,  at  81  per  cent,  per  annum? 

Ans.  $111.91f 

6.  What  is  the  interest  of  $5430  for  17  months,  at  4 
per  cent  per  annum?  Ans.  $307.13i. 

7.  What  is  the  interest  of  $7200  for  14  months,  at  6 
percent,  per  annum?  Ans.  $$04. 

8.  What  is  the  interest  of  $8050.871  for  3  years  and 
11  months,  at  6  per  cent,  per  annum? 

Ans.  $1891. 95c.  5m. 

9.  What  is  the  interest  of  $948.621  for  8  months,  at 
8  percent,  per  annum?  Ans.  $50.59c.  3m. 

10.  What  is  the  interest  of  £421.  16s.  9d.  for  2  years 
and  8  months,  at  5  per  cent,  per  annum? 

Ans.  £56.  4s.  lOfd. 

11.  What  is  the  interest  of  580  pounds  for  5  years,  2 
months  and  10  days,  at  7  per  cent  per  annum? 

Ans.  £210. 17s. 

12.  What  is  the  interest  of  $36  for  1  month,  at  8  per 
cent,  per  annum?  Ans.  24  cents. 

When  the  rate  is  6  per  cent,  another  method  ^of  finding 
the  interest  for  any  number  of  months,  is,  to  multiply  the 
principal  by  half  the  number  of  months  and  divide  the  pro- 
duct by  100  (or  cut  off  the  two  right  hand  figure*  as  before.' 

KLUIPLES. 

1.  What  is  the  interest  of  $1500  for  4  months? 
$1500 

2  half  the  number  of  months.  - 

30(00  Ans.  $30. 


SIMPLE   INTEREST.  135 

2.  What  is  the  interest  of  $7656  for  3  years  and  4 
months?  $7656 

20    half  the  number  of  months. 

1531J20  Ans.  $1531.20. 

3.  What  is  the  interest  of  $230.25  for  8  months? 

Ans.  $9.21. 

4.  What  is  the  interest  of  $750  for  9  months? 

Ans.  $33.75. 

5.  What  is  the  interest  of  $967.64  for  28  months? 

Ans.  $135.47. 

The  interest  for  any  number  of  days*  at  6  per  cent,  can 
be  found  by  multiplying  the  dollars  by  the  number  of  days, 
&id  dividing'  the  product  by  83:  the  answer  will  be  in  cents, 
if  the  principal  consist  (f  dollars  and  cents ,  cut  ojf  the  two 
right  hand  figures.  \gj~Bank  interest  is  reckoned  by  this 
rule.  • 

1.  What  is  the  interest  of  $1542  for  90  days?    And 

of  $754.54  for  60  days? 

$1542  $754.54 

90  60 


6|0)13S7810  6|0)452724|0 

2313  Ans.  $23.13.  754|54  Ans.  $7.54. 

2.  What  is  the  interest  of  $3084  for  30  days  at  6  per 
cent,  per  annum?  Ans.  $15.42. 

3.  What  is  the  interest  of  $2324  for  54  days,  at  6  per 
cent,  per  annum?  Ans.  $20.91. 

4.  What  is  the  interest  of  $281.75  for  93  days,  at  6 
per  cent,  per  annum  ?  Ans.  $4.36. 

CASE  III. 

The  amount,  time,  and  rate  per  cent,  given  to  find  the 
principal 

RULE. 

Find  the  amount  of  100  dollars  at  the  rate  and  time 
given;"  then  say,  as  the  amount  of  100  dollars,  is  to  the 
amount  given,  so  are  100  dollars  to  the  principal  re- 
quired.. 


SIMPLE    INTEREST. 
EXAMPLES. 


. 

1.  What  principal  at  interest  for  two  years,  at  6  per 
;ent.  per  annum,  will  amount  to  $134.40? 

iifrl  HA 


$100 
6 


12.00 
100.00 

$112  amount  of  100  for  two  years. 
dolls.        $  cts.        dolls,    dolls. 
Then,  as  112   :  134.40     :  ;   100  -    120  Ans. 

2.  What  principal  at  interest  for  5  years,  at  6  per 
sent,  will  amount  to  $780  ?  Ans.  $600. 

3.  What  principal  at  interest  for  4  years  and  3  months, 
it  6  per  cent,  will  amount  to  $119235?     Ans.  $950. 

CASE  IV. 

Tofnd  the  rate  per  cent,  when  ike  amount^  time  and  prin- 
cipal are  given. 

KULE. 

Take  the  principal  from  the  amount,  the  remainder 
will  be  the  interest  for  the  given  time;  then,. as  the  prin 
cipal  is  to  one  hundred  dollars,  so  is  the  interest  of  the 
principal  for  the  given  time,  to  the  interest  of  100  dollars 
for  the  same  time.  Divide  the  interest  of  100  dollars 
thus  found,  by  the  time,  and  the  quotient  will  be  the 
rate  per  cent. 

EXAMPLES. 

1.  At  what  rate  per  cent,  will  $500  amount  to  $650 
in  three  years  ?  650  Amount. 

500  Principal. 

150  Interest  for  the  time. 
D.        D.          D.      D. 
As  500   :  100   :   :  150  .  30  Interest  of  100. 
Then  divide  by  the  time  3)30(10  Ans.  per  cent, 
30 


COMPOUND    INTEHK3T,  137 

2-.  At  what  rate  per  cent,  per  annum  will  $1850  dou- 
ble in  5  years?  Ans.  20  per  cent. 

3.  At  what  rate  per  cent,  per  annum,  will  600  dol- 
lars amount  to  $856.50  in  9  years  and  6  months? 

Ans.  4£  per  cent. 
CASE  V. 

To  find  the  time  when  the  principal,  amount,  and  rale 
per  cent,  are  given. 

RULE. 

Find  the  interest  of  the  principal  for  one  year;  find 
the  interest  of  the  principal  for  the  whole  time,  by  sub- 
tracting the  principal  from  the  amount;  then  divide  the 
whole  interest  by  the  interest  for  one  year — the  quotient 
will  show  the  time  required. 

EXAMPLES 

1.  In  what  time  will  $800  amount  to  $1000  at  5  per 
cent. per  annum? 

800  1000  Then,  4|0)20|0 

5  800  5 

$40|00  200  Whole  In't.  Ans.  5  years. 

2.  In  what  time  will  $80  amount  to  $182.40  at  8 
per  cent,  per  annum?  Ans.  16  years. 

3.  In  what  time  will  $600  amount  to  $798  at  6  pei 
cent,  yer  annum?  Ans.  5£  years. 

'  COMPOUND  INTEREST. 

Compound  Interest  is  that  which  arises  from  the  in 
terest  being  added  to  the  principal,  and  becoming  a  par 
of  the  principal,  at  each  time  of  payment. 

RULE. 

Find  the  amount  of  the  principal,  for  the  time  of  the 
first  payment,  by  Simple  Interest;  this  amount,  contain 


38 


COMPOUND    INTEREST. 


EXAMPLES. 

1.  What  is  the  compound  interest  of  $8000  for  two 
-ears,  at  6  per  cent  per  annum  ? 


Interest  for  the  first  year  480|00 
Principal  8000 


Amount  8480 
6 

In't.  for  the  second  year  508.|80 
Principal  8480.00 

8988.80 
Subtract  8000.00 


$988.80c.  Answer. 

2.  What  is  the  compound  interest  of  $554  for  3  years, 
at  8  per  cent,  per  annum?  Ans.  $143.88. 

3.  What  is  the  compound  interest  of  $744  for  2  years, 
at  7  percent,  per  annum?  Ans.  $l07.80c.  5m 

4.  What  is  the  compound  interest  of  $50  for  8  years, 
at  8  per  cent,  per  annum?  Ans.  $42.54c.  6m 

5.  What  is  the  compound  interest  of  £48.  5s.  for  3 
years,  at  6  per  cent,  per  annum?    Ans.  £9.  4s.  3|xl. 

In  computing  Interest  on  Notes. 

When  a  settlement  is  made  within  a  short  time  from  the  date 
or  commencement  of  interest,  it  is  generally  the  custom  to  pro- 
ceed according  to  the  following 

RULE. 

Find  the  amount  of  the  principal,  from  the  time  the  inte 
rett  commenced  to  the  time  of  settlement,  and  likewise  the 
amount  of  each  payment,  from  the  time  it  was  paid  to  the 
time  of  settlement;  then  deduct  the  amount  of  the  sever  a 
payments  from  the  amount  of  the  principal. 
Exercises  for  the  Slate. 

1.  For  ralue  received,  I  promise  to  pay  Rufus  Stanly,  or  order 
Three  Hundred  Dollars,  with  interest.  April  1, 1825. 
$300.  PETER  MOSELY. 


INTEREST. 


139 


On  tnis  note  were  the  following  endorsements : — 
October  1, 1825,  received  $100  > 
April  16,  1826,     .     .     .    $  50V 
December  1,  1827,    .     .    $120} 
tf  hat  was  due  April  1,  1828?        Ans.  $60.73, 

CALCULATION, 

The  first  principal  on  interest  from  April  1, 1825,  -    $300.00 
Interest  to  April  1,  1828,  (36  mo.),       -        -        -        54-00 


Amount  of  principal 
?irst  payment,  Oct.  1, 1825>     - 
•nterest  to  April  1, 1828",  (30  mo.) 
Second  payment,  April  16,  1826, 
Interest  to  April  1,  1828,  (23*  mo.) 
Third  payment,  Dec.  1,  1827,     - 
Interest  to  April  1, 1828,  (4  mo.} 


$100.00 

15.00 

50.00 

5.87 

120.00 

2.40 


-  $354.00 


Amount  of  payments  deducted  -  -   $293.27 

Remain*  due,  April  1, 1828,      -    -     $60.73 
2.  For  value  received,  I  promise  to  pay  Peter  Trusty,  or  order, 
Five  Hundred  Dollars,  with  interest.    July  1, 1825. 
$500.  JAMES  CARELESS. 

ENDORSEMENTS. 
July  16, 1826,  received  $200) 
Jan.  1,1827,    -    -    -    $  40V 
March  16,  1827,    -    -    $230) 
What  remained  due  July  16, 1828?        Ans.  $75.15. 

RULE  IN  SOME  OF  THE  UNITED  STATES. 

Compute  the  interest  on  the  principal  sum  to  the  first  time 
when  a  payment  was  made,  which,  either  alone,  or  together 
with  the  preceding  payments  (if  any,)  exceeds  the  interest 
then  due;  add  that  interest  to  the  principal,  and  from  the  sum 
subtract  the  payment,  or  the  sum  of  the  payments,  made 
that  time,  and  the  remainder  will  be  a  new  principal,  with 
which  proceed  as  with  the  Jirst  principal,  and  so  on,  to  the 
time  of  settlement. 

1.  For  value  received,  I  promise  to  pay  Jason  Park,  or  order 
Six  Hundred  Dollars,  with  interest.     March  1, 1822. 

$600.  STEPHEN  STIMPSON. 

ENDORSEMENTS. 
May    1, 1823,  received  $2001 
June  16,  1824,        -         $  80 
Sept.  17, 1825,        -         $  12  | 
Dec.  19, 1825,       -         $  15  I 
March  1,1826,        -         $100 
Oct.  16,  1827,        -         $150  J 
What  was  there  due  August  31,  1828?    Ans.  $194.41. 


140  INTEREST. 

The  principal,  $600,  on  interest  from  March  1,  1822,       $600.00 
Interest  to  May  1, 1823,  (14  mo.)        - 


Amount,  $642.00 
Payment,  May  1,  1823,  a  sum  greater  than  the  interest,        200.00 


Due  May  1,  1823,  forming  anew  principal,        -        -      $442.00 
Interest  on  $442,  from  May  1,  1823,  to  June  16,  1824, 

(13*  mo.)        -        -       -        -        -        -        -  29.83 

Amount,  $471.83 
Payment,  June  16,  1824,  a  sum  greater  tlian  the  interest 

then  due, 80.00 


Due  June  16,  182$,  forming  a  new  principal,        -        -     $391.83 
Interest  on  $391.83,  from  June  16,  1824,  to  March  1, 

1826,  (20ft  mo.) 40.16 


Amount,  $4ol.99 

Payment,  a  sum  less  than  the  interest  then  due,  $  12 
Payment,  a  sum  less  than  the  interest  then  due,  $  15 
Payment,  a  sum  greater  than  the  interest  then  due,  $100 

$127.00 

Due  March  1,  1B26,  forming  a  new  principal,        -  $304.99 

Interest  on  $304.99,/rom  March  1, 1826?fo  Oct.  16, 1827, 

(19  i  mo.)  -        -        -•  _29.73 

Amount,  $334.7 
Payment,  Oct.  16, 1827,  a  sum  greater  th&n  the  interest 

then  due,        -        -        -    '   -        -        -       -        -        150.00 

Due  Oct.  16, 1827,  jorming  a  new  principal,       -        -     $184.72 
Interest  on  $184.72,  from  Oct.  16,  1627,  to  August  31, 

1828,  being  the  time  of  settlement,  (10i  mo.)     -        -  9.69 


Balance  due  Aug.  31, 1828,        -  $194.41 

2.  For  value  received,  I  promise  to  pay  Asher  L.  Smith,  or 
order,  Nine  Hundred  Dollars,  with  interest.    June  16, 1820. 
$900.  WILLIAM  MoRRre. 

ENDORSEMENTS. 
July    1,  1821,  received  $150 
Sept.  16,  1822,  -  -     -     $  90 


-  $10  | 

-  $20) 

-  fee 


Dec.  10,  1824,  -  - 
June  1,  1825,  -  - 
Aug.  16,  1825,  -  - 
March  1,  1827,  -  - 
What  remained  due  Sept.  1, 1828?  Ans.  '$483.07. 

Q.  1.  What  is  Interest? 

2.  What  are  the  four  things  considered  in  calculating 

interest? 
o.  What  is  the  principal  ? — What  is  the  rate  per  cent 

What  is  the  time? — What  is  the  amount? 


COMMISSION    AND    BliOKEKAGK.          14 j 

4.  How  do  you  proceed  in  the  first  case? 

5.  How  do  you  proceed  in  pounds,  shillings,  &c.? 

6.  How  do  you  proceed  when  the  rate  per  cent,  con- 

sists of  a  whole  number  and  a  fraction f 

7.  How  do  you  proceed  when  the  principal  is  in  dol- 

lars and  cents? 

8.  How  do  you  calculate  interest  for  more  than  a 

year? — How,  when  the  time  is  in  mouths? 

9.  What  other  method  is  there  for  calculating  interest, 

besides  the  method  of  multiplying  the  sum  hy 
the  rate  per  cent.? 

10.  How  is  bank  interest  reckoned? — What  is  the  rule 

for  casting  it? 

11.  Do  you  understand  all  the  cases  and  rules  of  in- 

terest? 

12.  What  is  Compound  Interest? 

13.  Repeat  the  rule  for  calculating  Compound  Interest? 


INSURANCE,  COMMISSION  AND  BROKERAGE. 

Insurance,  Commission  and  Brokerage,  are  premiums 
allowed  to  insurers,  factors  and  brokers  at  a  certain  rate 
per  cent.;  and  is  obtained  after  the  manner  of  the  first 
case  in  Simple  Interest. 

EXAMPLES. 

1.  What  is  the  insurance  of  $4500,  at  2i  per  cent.? 

•-J? 

9000 
2250 

$112i50c.  Answer, 

2,  What  is  the  commission  on  a  sale  of  goods  amount- 
ing to  $1184  at  5  per  cent.?  Ans.  §59.20. 

3,  What  is  the  brokerage  of  $987  at  3  per  cent.? 

Aris.  $-29.61. 

4.  What  is  tho  commission  on  a  sale  of  goods  amount- 
ing to  4820  at  4J  per  cent,?  $216.90. 


142  DISCOUNT. 


DISCOUNT. 

Discount  is  an  allowance  made  for  the  payment  of 
any  sum  of  money  before  it  becomes  due,  and  is  the  dif- 
ference between  that  sum,  due  some  lime  hence,  and  its 
present  worth. 


As  the  amount  of  $100  at  the  given  rate  and  time  is 
to  $100,  so  is  the  given  sum  or  debt  to  the  present  worth. 
Subtract  the  present  worth  from  the  given  sum,  and  the 
remainder  will  be  the  discount. 

EXAMPLES. 

1.  What  is  the  present  worth  of  $500  due  in  3  years, 
it  6  per  cent,  per  annum? 

8(2  <2» 

tp  w 

$•100  113    :  ICO   :    :  500 

6  100 

6|00  118)50000(423.72. 

3  472 


18  280 

100  236 


118  amount  of  $100.        440 

354 

S6.00|72c. 
826 

340 
236 

104  remainder. 

2.  What  is  the  present  worth  of  $350  payable   in  6 
months,  discounting  at  6  per  cent,  per  annum? 

Ans.  $339.80c.  5m. 

3.  What  is  the  discount  on  01000  due  in  one  year, 
at  6  per  cent,  per  annum?  Ans.  $56.60c.  4m. 

4.  What  is  the  present  worth  of  £65  due  in  15  months 
at  6  per  cent,  per  annum?  Ans.  £60.  9s.  3£d. 


EQUATION.  143 

5.  What  sum  will  discharge  a  debt  of  $1595  due  af- 
ter 5  months  and  20  days  at  6  percent,  per  annum? 

Ans.  $1541.32.  6m. 

6.  What  is  the  present  worth  of  $426.55  at  6  per  cent, 
per  annum,  due  in  8  months?        Ans.  $410. 14c.  5m. 

Note. — When  discount  is  made  without  regard  to  time, 
it  is  found  as  the  interest  of  the  sum  would  be  for  one 
year. 

EaUATION. 

Equation  is  the  method  for  finding  a  time  to  pay  at 
once,  several  debts  due  at  different  times. 

RULE. 

Multiply  each  payment  by  the  time  at  which  it  is  due, 
and  divide  the  sum  of  the  products  by  the  sum  of  all  the 
payments — the-  quotient  will  be  the  time  required. 

EXAMPLES. 

1.  A  owes  B  $480  to  be  paid  in  the  following  man- 
ner, viz:  $100  in  6  months,  $120  in  7  months,  and 
$260  in  10  months;  what  is  the  equated  time  for  pay- 
ment of  the  whole  debt? 

100  X  6=  600 

120  X  7=  840 

260x10=2600 


480  )4040(8,V  months,  Ans. 

3840 

200 


2.  A  owes  B  $1100,  of  which  200  is  to  be  paid  in  3 
[months,  400  in  5  months,  and  500  in  8  months  —  what 
jis  the  equated  time  for  payment  of  all?     Ans.  6  months. 

3.  C  is  indebted  to  a  merchant  to  the  amount  of  $2500; 
of  which  $1000  is  payable  at  the  end  of  4  months,  $800 
in  8  months,  and  700  in  12  months  —  when  ought  pay- 
ment to  be  made,  if  all  are  paid  together? 

Ans.  7  months,  153  days. 


144  LOSS    AND    GAIN, 

LOSS  AND  GAIN. 

Loss  and  Gain  is  a  rule  by  which  persons  in  trade  are 
able  to  discover  their  profit  or  loss ;  and  to  increase  or 
lessen  the  prices  of  their  goods  so  as  to  gain  or  lose  on 
them  to  any  given  amount. 

Questions  in  Loss  and  Grain  are  solved  by  the  Rule 
of  Three,  or  by  Practice. 

EXAMPLES. 

1.  A  merchant  bought  100  yards  of  silk  at  75  cents 
per  yard,  what  will  be  his  gain  in  the  sale,  if  he  sell  it 
for  90  cents  y er  yard  ? 

75  cents. 

yard,  yards.       ctx.     dolls. 
15  gain  per  yard.  As  1  :    100   :  ;    15   ;    15  Ans. 

2.  If  a  grocer  buy  250  Ibs.  of  tea,  at  $225,  and  sell 
the  whole  at  $1.25  per  Ib.  what  will  be  his  gain  by  the 
transaction?  Ans.  $87.50. 

3.  If  a  yard  of  calico  cost  28  cents,  and  is  sold  for  31 
cents,  what  is  the  gain  on  293  yards?  Ans.  $8.79. 

4.  Bought  420  bushels  of  corn  at  25  cents  per  bushel, 
and  sold  the  same  at  38  cents  per  bushel;  what  was  the 
amount  gained?  Ans.  $54.60. 

5.  A  merchant  bought  12  cwt.  of  coffee  at  26  cents 
per  Ib.  and  afterwards  was  obliged  to  sell  it  at  20  cents 
per  Ib.  what  was  his  loss?  Ans.  $80.64, 

6.  If  a  merchant  gain  $80  on  $560,  what  is  that  per 
cent.?  Ans.  14^  per  cent. 

7.  If  a  yard  of  velvet  be  bought  for  16s.  and  sold  again 
for  12s.  what  is  the  loss  per  cent.?       Ans.  25  per  cent. 

8.  A  merchant  bought  2  hhds.  of  wine,  containing 
126  gals.,  at  $1.75  a  gal.  and  retailed  the  same  at 
$2.12 J  a  gal.:  what  did  he  gain  in  the  whole? 

Ans.  $47.25. 

9.  A  merchant  bought  2  pieces  of  broad-cloth,  con- 
taining 56  yds.,  at  $4.75  a  yard ;  but  upon  examination, 
found  them  damaged.     He  was,  therefore,  obliged  to  sell 
them  for  $4.12  J  a  yard;  how  much  did  he  lose  by  the 
bargain?  Ans.  $35. 


INVOLUTION  145 

10.  A   gentleman   purchased   1500  Ibs.  of  coffee  for 
$172.50,  how  must  he  sell  the  same  to  gain  $32  b\  his 
bargain?  Ans.   13  cts.  6  in.  33. 

11.  A  merchant  bought  250  hbls.  of  flour  ;M  g: 
bbl.;  how  must  he  sell   the  same  to  gain   $55  b: 
bargain?  An-.  :     .     . 

12.  A  lady  purchased  a  quantity  of  milliner} ,  r'-r  u  hich 
she  gnve  $184;  and  sold  the  same  for  $2!0;  how  m.srn 
did  she  gain  per  cent.?  Ans.  14.13-f-per  cent. 


INVOLUTION, 

OR  THE  RAISING  OF  POWERS. 

The  product  of  any  number  multiplied  by  itself  any 
given  number  of  times,  is  called  its  power,  as  in  the  f  >I- 
lowing  example. 

Thus,     2x2=4  the  square,  or  second  power  of  2. 

2x2x2=S  the  cube,  or  third  power  of  2. 
2x2x2x2  =  16*  the  liqundrate,  or  fourth  power  of 
2.*     Hence,  3  r  used  to  the  4th  power  mnkes  81.     The 
number  which  denotes  a  power  is  called  the  index,  or 
exponent  of  that  power. 

When  a  power  of  a  vulgar  fraction  is  required,  it  is 
only  necessary  to  raise,  first  the  numerator,  and  then  the 
denominator  to  the  given  power,  and  place  the  product 
of  the  one  over  the  product  of  the  other,  thus,  the  3d 

power  of  J  <jv3  y  3=±^V 

EXAMPLES. 

1.  What  is  the  square  of  4567?  Ans.  20857489. 

2.  What  is  the  cube  of  567?  Ans.  182284263. 

3.  What  is  the  biquadrate  of  67'        Ans    20151121. 

4.  What  is  the  ninth  power  of  2?  Ans.  512. 

5.  What,  is  the  cube  of  J?  Ans.   ?|£. 

6.  What  is  the  cube  or  third  power  of  ,13? 

Ans.  002197 

7.  What  is  the  sixth. power  of  5.03? 

Ans.   16196.005304471)720. 


*Any  eriven  number  is  co-  siderod  the  first  power  of  itself,  and 
when  multiplied  by  itself  th    product  is  the  second  power,  &c : 


14(3  EVOLUTION. 

EVOLUTION, 

OR  THE  EXTRACTION  OF  ROOTS. 

The  root  of  a  number,  or  power,  is  any  number,  which 
being  multiplied  by  itself  a  certain  number  of  times, 
will  produce  that  power;  and  is  called  the  square,  cube, 
biquadrate  root,  &.c.  according  to  the  power  to  which  it 
belongs.  Thus,  3  is  the  square  root  of  9,  because  when 
multiplied  by  itself,  it  produces  9;  and  4  is  the  cube  root 
of  (34,  because  4x4x4=64*  and  so  of  any  other  num- 
ber. 

THE  SQUARE  ROOT. 

Extracting  the  square  root  of  a  number,  is  the  taking 
a  smaller  number  from  a  larger,  and  such  as  will,  being 
multiplied  by  itself,  produce  the  larger  number. 

RULE. 

1.  Separate  the  sum  into  periods  of  two  figures  each 
beginning  at  the  right  hand  figure. 

5.  Seek  the  greatest  square  number  in  the  left  liand 
period;  place  the  square,  thus  found,  under  that  period, 
and  the  root  of  it  in  the  quotient.  Subtract  the  square 
number  from  the  first  period;  to  the  remainder  bring 
down  the  next  period,  and  call  that  the  resoivend. 

3.  Djubie  the  quotient,  and  place  it  on  the  left  hand 
of  the  resolverid  for  a  divisor.  Seek  how  often  the  divi- 
sor is  contained  ki  the  resoivend,  omitting  the  units  fig- 
ure, and  set  the  answer  in  the  quotient,  and  also  on  the 
right  hand  side  of  the  divisor.  Then  multiply  the  divi- 
sor, including  the  last  added  figure,  by  that  figure,  that 
is,  by  the  figure  last  placed  in  the  quotient;  place  the 
product  under  the  resoivend,  subtract  it,  and  to  the  re- 
mainder bring  down  the  next  period,  if  there  be  any  more, 
and  proceed  as  alreidy  directed.  If  there  be  a  remain- 
der after  the  periods  are  all  brought  down,  annex  cyphers, 
|t.wo  at  a  time,  tor  decimals,  and  proieed  till  the  root  is 
;obtained  with  sufficient  exactness. 

Note. — When  a  sum  in  the  Square  Root  consists  of 
whole  numbers  and  decimals,  point  off  the  whole  num- 
bers as  above  directed,  then  point  the  decimal  part, 


EVOLUTION.  147 

Commencing  at  the  decimal  point  and  forming  periods 
of  two  figures  each  towards  the  right,  observing  when 
there  is  only  one  figure  left  for  the  last  period,  to  add  u 
cypher  to  the  right  of  it,  to  make  an  even  period. — 
When  the  sum  consists  entirely  of  decimals,  separate 
the  periods  after  the  same  manner.  If  it  le  required  to 
extract  the  square  root  of  a  vulgar  fraction,  reduce  it  to 
its  lowest  terms;  then  extract  the  root  of  the  numerator 
[for  the  numerator  of  the  answer,  and  the  root  of  the  de- 
nominator for  the  denominator  of  the  answer.  If  the 
fraction  be  a  surd,  that  is,  a  number  u  hose  root  can  never 
be  exactly  found,  reduce  it  to  a  decimal,  and  then  ex- 
tract the  root  from  it;  and  if  the  sum  1  e  a  mixed  number, 
the  root  may  be  obtained  in  tho  same  way. 

PROOF. 

Square  the  root,  adding  the  remainder,  (if  any,)  and 
the  result  will  equal  the  given  number. 

EXAMPLES. 

1.  What  is  the  square  root  of  20857489? 

....  Root. 
20857481(1567  Answer. 
16 

divisor  85)185  resolvend. 
425 


divisor  906)3074  resolvend. 
5436 

divisor  9127)  >3S89  resolvend. 
63889 


2.  What  is  the  square  root  of  294849?         Ans.  543 

3.  What  is  the  square  root  of  41242084?  A n?.  6422 

4.  Wh  t  is  the  sq-uire  root  of  17.3056?       AriF.  4.H: 

5.  What  is  the  sq  lare  root  of  .00072k?       Ans.  .027 

6.  Whet  is  the  sq  lare  root  of  5?  Ans.  2.23600 

7.  What  is  the  square  r:>ot  of  T\7T?  An.«.  if 

8.  What  is  the  square  root  of  1  /£?      Ans.  4.168333 


148  EVOLUTION. 

9.  A  general  has  an  army  of  7056  men;  how  many 
must  he  place  on  a  side  to  form  them  into  a  compact 
quare?  Ana.  84. 

10.  If  the  area  of  a  circle  be  184.125,  what  is  the  side 
>f  a  square  that  shall  contain  the  same  area? 

Thus,  «/184.125=13.569+Answer. 
11.  If  a  square  piece  of  land  contain  61  acres  and  41 
square  poles,  what  is  the  length  of  one  of  its  sides? 

A.  P. 

Thus,  61  41=9801  square  poles. 
Then,  ,J 380 1=99  rods,  or  poles  in  length,  Answer^ 

12.  There  is  a  circle  whose  diameter  is  4  inches;  what 
is  the  diameter  of  a  circle  3  times  as  large? 

Thus,  4X4=16;  and  16X3=48  and 
-[-inches.     Ans. 

13.  There  is  a  circle  whose  diameter  is  8  inches;  what 
is  the  diameter  of  a  circle  which  is  only  one  fourth  as 
arge.     8x8=64;  and 64-^-4= 16;  and  ,yi6=4inches. 

Aas.  4  inches. 

The  square  of  the  longest  side  of  a  right  aug'edtriarigle, 
is  equal  to  the  sum  of  the  squares  of  the  other  two  sides; 
therefore,  the  difference  of  the  squares  of  the.  longest  side, 
and  either  of  the  other  sides ,  is  the  square  of  the  remaining 
side. 

14.  The  wall  of  a  certain  city  is  20  feet  in  height,  it  is 
surrounded  by  a  ditch  20  feet  in  breadth;  what  must  be 
the  length  of  a  ladder,  to  reach  from  the  outside  of  the 
ditch  to  the  top  of  the  wall?  Ans.  28£  feet. 

15.  On  the  margin  of  a  river  24  yards  wide,  stands  a 
tree;  from  the  top  of  which  a  line  36  yards  long,  will 
reach  to  the  other  side  of  the  stream;  what  is  the  height 
•jf  the  tree  ?  Ans.  26.83-j-  yards. 

16.  Two  ships  sail  from  the  same  port;  one  sails  due 
e;ist  50  miles,  and  the  other  due  south  84  miles;  how 
far  are  they  from  each  other?          Ans.  97.75  miles. 

17.  A  ladder  or  pole,  40  feet  long,  placed  in  the  middle 
of  a  street,  will  reach  a  window  of  a  house  on  each  side 
of  the  street  24  feet   from  the   pavement;  what  is  the 
width  of  the  street?  Ans.  64  feet  wide. 


THE  CUBE   KOOT.  149 

THE  CUBE  ROOT. 

The  cube  root  of  a  given  number,  is  such  a  number 
as  being  multiplied  by  itself,  and  then  into  that  product, 
produces  the  given  number. 

RULE, 

1 .  Point  off  the  sum  into  periods  of  three  figures  each, 
[beginning  with  units. 

2.  Find   the  greatest  cube  in  the   left  hand  period, 
place  the  root  of  it  in  the  quotient,  subtract  the  cube 
[from  the  left  hand  period,  and  to  the   remainder  bring 

own  the  next  period  for  a  resolvend. 

3.  Square  the  quotient  and  multiply  the  square  by  3 
for  a  defective  divisor. 

4.  Seek  how  often  the  defective  divisor  is  contained 
the  resolvend,  omitting  the  units  and  tens,  or  two 

right  hand  figures.  Place  the  result  in  the  quotient, 
and  its  square  to  the  rightof  the  divisor,  supplying  the 
place  of  tens  with  a  cypher,  whenever  the  square  is  less 
than  ten. 

5.  Multiply  the  last  figure  of  the  quotient  or  root  by 
all  the  figures  in  it  previously  ascertained;  multiply  that 
product  by  30,  and  add  their  product  to  the  divisor,  to 
complete  it. 

6.  Multiply  and  subtract  as  in  Simple  Division,  and 
to  the  remainder  bring  down  the  next  period,  for  a  new 
resolvend.     Find  a  divisor  as  before,  and  thus  proceed 
until  all  the  periods  are  brought  down. 

Note. — The  cube  root  of  a  vulgar  fraction  is  found  by 
jreducing  it  to  its  lowest  terms,  and  extracting,  as  in  the 
square  root;  and  if  the  fraction  be  a  surd,  reduce  it  to  a 
(decimal,  and  then  extract  the  root. 

In  extracting  the  cube  root,  if  the  sum  be  in  part  de- 
|cimals,  or  if  the  whole  be  decimals,  point  the  figures 
in  the  square  root,  observing  to  have  three  figures  in  a 
period  instead  of  t\vo;  and  in  all  cases  in  the  cul  e  root, 
when  there  is  a  remainder,  if  it  be  required  to  obtain 
decimal  figures  to  the  root,  proceed  as  directed  in  the 
»qna~e  root,  only  add  three  cyphers,  in  place  of  two  to 
'the  remainder. 


150  THE    CUBE    ROOT. 

PROOF. 

Involve  the  root  to  the  third  power,  adding  the  remain 
der,  (if  any,)  to  the  result. 

EXAMPLES. 

1.  What  is  the  cuoe  root  of  182284263? 

.      .     .  Root. 

Uefec.divJ   ,     ,       «       753G  182284263(567 

&sq.of6.j  5X5X  t3-  7jl  125 

6X5X30=    900 

Complete  divisor — 6436  )57284  rcsolv. 

50616 

56X56X3  =940849          

7x56x30=   11760 

Complete  divisor  952609"  )6668263 

6668263 

2.  What  is  the  cube  root  of  48228.544?      Ans.  36.4. 

3.  What  is  the  cube  root  (or  3d  root)  of  2? 

Ans.  1.259921. 

4.  What  is  the  cube  root  of  132651?  Ans.  5J. 

5.  What  is  the  cube  root  of  4173281  ?          Ans.  161. 

6.  What  is  the  cube  root  of  .008649?   Ans    .2052-}-. 

7.  What  is  the  cube  root  of  iff?  Ans. 

8  What  will  be  the  cube  root  of  160,  the  decimal  be- 
ing continued  to  three  places?  Ans.  5.428-}-. 

9.  If  the  contents  of  a  globe  amount  to  5832  cubick 
inches,  what  are  the  dimensions  of  the  side  of  a  cubick 
block  containing  the  same  quantity?  Ans.  18  in.  square. 

10.  If  the  diameter  of  the  planet  Jupiter  is  12  times  as 
I  much  as  the  diameter  of  the  earth,  how  many  globes  of 
the  earth  would  it  take  to  make  one  as  large  as  Jupiter? 

Ans.  1728. 

11.  If  the  suu  is  1000000  times  as  large  as  the  earth, 
I  and  the  earth  is  8000  miles  in  diameter,  what  is  the  di- 
ameter of  the  sun?  Ans.  8DOOOO  miles. 

Note. — The  roots  of  most  powers  may  be  found  by  the 
square  and  cube  roots  only;  thus  the  square  root  of  the 
square  root  is  the  biquadrate,  or  4th  root,  and  the  sixth 
root  is  the  cube  of  this  square  root. 


ALLIGATION.  151 

Questions  concerning  the  powers  and  roots. 

1.  What  is  called  a  power? 

2.  What  power  is  the  square?      Ans.  The  2d  power. 

3.  What  is  the  cube  of  a  number  called? 

4.  How  do  you  raise  the  power  of  a  vulgar  fraction? 

5.  What  is  the  root  of  a  power? 

6.  What  is  meant  by  extracting  the  square  root? 

7.  Repeat  the  rule  for  doing  it. 

8.  How  do  you  proceed  when  the  sum  consists  in  part, 

or  altogether,  of  decimals? 

9.  How  do  you  extract  the  root  of  a  vulgar  fraction  ? 

10.  How  do  you  proceed  when  the  fraction  is  a  surd? 

11.  What  dc  you  understand  by  the  cube  root? 

12.  Repeat  the  rule  for  extracting  it. 

13.  How  are  sums  in  the  square  root  proved? 

14.  How  are  sums  in  the  cube  root  proved? 


ALLIGATION. 

Alligation  is  a  rule  for  mixing  simples  of  different 
qualities,  in  such  a  manner  that  the  composition  may  be 
of  a  meaner  middle  quality. 

CASE  I. 

To  fold  the  mean  price  of  any  part  of  the  mixture,  wJien 
the  quantities  and  prices  of  several  things  are  given. 

RULE. 

As  the  sum  of  the  quantities  is  to  any  part  of  the  com- 
position, so  is  the  price  of  the  quantities  to  the  price  of 
any  particular  part. 

EXAMPLES. 

1.  A  trader  mixes  60  gallons  of  wine  at  100  cents  per 
gallon ;  40  gallons,  at  80  cents,-  and  30  gallons  of  water. 
What  should  be  the  price  per  gallon? 
gals.        cte.      $ 

Wine    60  at  100=60,00 

Wine    40  at    80=32.00 

Water  30  

gals.       gal.      $ 
130     :     1  :  :  92.00.     Ans.  70 . 


152  ALLIGATION. 

2.  A  trader  mixes  a  quantity  of  tea  as  follows,  viz:  — 
4  Ibs.  of  tea  at  42  cents  per  lb.;  6  ibs.  at  33  cents;  12 
Ibs.  at  75  cents,  and  15  ibs.  at  80  ceMts.     What  can  he 
sell  it  for  per  lb.?  Ans.  6b'JA  cents. 

3.  A  farmer  mixes  20  bushels  of  wheat  at  5s.  per 
mshel,  with  33  bushels  of  rye  at  3s  ,  and  40  bushels  of 
>arley  at  2s,  per  r/ushel  ;  how  much  is  a  bushel  of  the 
nixture  worth?  Ans.  3s. 

CASE  II. 

WJicn  the  prices  of  several  simples  are  given  to  find  what 
quantity  ofeach^  at  their  respective  pr  ices  <must  be  taken 
to  maJce  a  compound  at  a  proposed  price. 

HULE. 

Set  the  prices  of  the  simples  in  a  column  under  each 
>ther.  Connect  with  a  continued  line,  the  rate  of  each 
iinple  which  is  less  than  that  of  the  compound,  with  one 
»r  any  number  of  those  that  are  greater  than  the  com 
pound,  and  each  greater  rate,  with  one  or  more  of  the 
ess.  Place  the  difference  between  the  mixture  rate, 
md  that  of  each  of  the  simples,  opposite  to  the  rates, 
.viih  which  they  are  linked.  Then,  if  only  one  differ 
ince  stand  against  any  rate,  it  will  be  the  quantity  be- 
,onging  to  that  rate;  but  if  there  be  several,  their  sum 
will  be  the  quantity.  Different  modes  of  linking,  will 
produce  different  answers. 

EXAMPLES. 

1.  A  merchant  would  mix  wines  at  17s.  18s.  and  22s. 
per  gallon,  so  that  the  mixture  may  be  worth  20s.  per 
gallon:  what  quantity  of  each  must  be  taken? 
^      2  at  17s. 


ras 

rate  20s.  3+2==5at22s 

Ans.  2  gallons  at  17s.,  2  at  18s.,  and  5  at  22s. 

2.  H^w   much  barley  at  40  cents,  corn  at  60,  and 

|wheat  at  80  cents  per  bushel,  must  he  mixed  together, 

that  the  compound  may  be  worth  62  J-  cents  per  bushel? 

Ins.    17i  bush,  of  barley,  17£of  corn,  and  25  of  wheat. 


ALLIGATION.  133 

CASE  III. 

When  the  price  of  all  the  simples,  the  quantity  of  one  of 
them,  and  the  mean  price  of  the  mixture,  are  given,  to 
Jind  the  quantities  of  the  other  simples. 


RULE. 

Find  an  answer  as  before,  by  connecting;  then,  as  the 
lifferenceof  the  same  denomination  with  the  giveriqunu- 
ity,  is  to  the  differences  respectively,  so  is  the  given 
luantity,  to  the  different  quantities  required. 

EXAMPLES. 

1.  How  much  gold  of  15,  17,  18,  and  22  carats  fine 
must  be  mixed  together  to  form  a  composition  of  40  oz. 
>f  20  carats  fine? 

fl5     ^  ...        2 

Mean  or  Mixture  J  17X  ...        3 

rate.  20        1  18M  •        -        -        2 

t22,yj  5+3+2=10 

Then  as  16   :     2   :   :  40   :     5J    A  10 

and  as  16   :  1 0  «   :  40   :  25  {  Auswer' 
Ans.  5  oz.  of  15, 17  and  18  carats  fine,  and  25  oz.  of 
22  carats  fine. 

2.  A  grocer  has  currents  at  4d.,  6d.,  9d.,  and  lid.,  per 
Ih.  and  he  would  make  a  mixture  of  240  Ibs.  that  migh: 
be  sold  at  8d.  per  ib.j  how  much  of  each  kind  must  he 
take? 

Ans.  72  Ibs.  at  4d.,  24  at  6d.,  48  at  9d.  and  96  at  1  Id. 

CASE  IV. 

When  the  prices  of  the  simples,  the  quantity  to  be  mixed, 
and  the  mean  price  are  given,  to  Jind  the  quantity  of 
each  simple. 

RULE. 

Connect  the  several  prices,  and  place  their  differences 
as  before;  then,  as  the  s:im  of  the  differences  thus  given, 
is  to  the  difference  of  e.ich  rate,  so  is  the  quantity  to  be 
compounded,  to  the  quantity  required. 


154  POSITION. 


EXAMPLES. 

How  much  sugar  at  9  cents,  11  cents,  and  14  cents 
per  Ib.  will  be  necessary  to  form  a  mixture  of  20  Ibs. 
worth  12  cents  per  ib.?  (9  1  2 

12  IHV  2 

14  J\ 


6 
Then,  as  8   :  2   :   :  20   :     5  Ibs.     9  cents.) 

8   :  2   :   :  20   :     5  Ibs.  11  cents.l  Answer. 
8   :  4   :   :  20   :  10  ll.s.  14  cents.) 

2.  A  grocer  has  sugar  at  24  cents  per  Ib.  and  at  13 
cents  per  Ib.;  and  he  wishes  so  to  mix  2cwt.  of  it,  that 
he  may  sell  it  at  16  cents  per  Ib.;  how  much  of  each 
Ikind  must  he  take?    Ans.  162  f£  Its.  of  that  at  13  cents, 
land  61-j*,-  Ibs.  of  that  at  24  cents. 

3.  How  many  gallons  of  water  must  be  mixed  with 
wine  worth  60  cents  per  gallon,  so  as  to  fill  a  vessel  of 
80  gallons,  that  may  be  sold  at41J  cents  per  gallon? 

Ans.  &>   gallons  of  water,  and  55  of  wine 

POSITION. 

Position  is  a  rule  for  solving  questions,  by  one  or  mor€ 
supposed  numbers.  It  is  divided  into  two  parts,  namel} 
single  and  double. 

SINGLE  POSITION. 

Single  position  teaches  to  solve  questions  which  re 
quire  but  one  supposition. 

RULE. 

Suppose  a  number,  and  proceed  with  it  as  if  it  were 
the  real  one,  setting  down  the  result — Then,  as  the  re 
suit  of  that  operation,  is  to  the  number  given,  so  is  the 
supposed  number,  to  the  number  sought. 

EXAMPLES. 

1.  What  number  is  that,  which  being  multiplied  by  7 
nd  the  product  divided  by  6,  will  give  14  for  the  quo 
tient?  Suppose  18 

6)726 
Then,  as  21    :  14   :   :  18   :  12  Answer. 


POSITION  155 

2.  What  number  is  that,  of  which  one  half  exceeds 
nc  third  by  15? 

"  Suppose  60— Then  i  |  60  |  J  J80 
30        *0 
Subtract  20 

To 

Then,  as  10   :  15   :   :  60   :  90  Answer. 

3.  What  number  is  that,  which  being  increased  by  1, 
and  i  of  itself,  the  sum  will  be  125?  Ans.  60. 

4.  A  schoolmaster  being  asked  how  many  scholars  he 
,ad,  answered,  that  if  |  of  his  number  were  multiplied 
>y  7,  and  J  of  the  same  number  added  to  the  product, 
he  sum  would  be  292.     What  was  his  number? 

Ans.  60. 

5.  A  schoolmaster  being  asked  what  number  of  schol- 
ars he  had,  said,  if  I  had  as  many,  half  as  many,   and 

>ne  fourth  as  many  more,  I  should  have  99.     What  was 
number?  Ans.  36. 

6.  A  person,  after  spending  1  and  J  of  his  money,  hao 
$30  left;  what  had  he  at  first?  Ans.  $180. 

7.  Seven  eighths  of  a  certain  number  exceed  four 
fifths  by  6.     What  is  that  number?  Ans.  80. 

8.  A  certain  sum  of  money  is  to  be  divided  among  < 
>ersons,  in  such  a  manner  that  the  first  shall  have  1  ol 
t,the  second  J,  the  third  £,  and  the  fourth  the  remain 

der,  which  is  $28;  what  is  the  sum?         Ans.  $112. 

9.  What  sum,  at  6  per  cent,  per  annum,  will  amoun 
3  £860,  in  12  years?  Ans.  £500. 

10.  A  person  having  about  him  a  certain  num1  er  oi 
crown*,  said,  if  a  third,  a  fourth  and  a  sixth  of  therr 
were  added  together,  the  sum  would  be  45;  how  man\ 
crowns  had  he?  Ans.  60. 

11.  What  is  the  age  of  a  person  who  says,  that  if -^  ~. 
the  years  he  has  lived  be  multiplied  by  7,  and  2  of  them 
be  a<dded  to  the  product,  the  sum  would  be29£  ? 

Ans.  60  years. 

12.  What  number  is  that,  which  being  multiplied  by 
and  product  divided  by  6,  the  quotient  will  be  14? 

Ans.  12. 


I5t>  POSITION. 

DOUBLE  POSITION. 

Double  Position  teaches  to  resolve  questions  by  means 
of  two  supposed  numbers. 

RULE. 

Suppose  two  convenient  numbers,  and  proceed  with 
each  according  to  the  condition  of  the  question,  and  set 
down  the  errours  of  the  results.  Multiply  the  errours 
'into  their  supposed  numbers,  crosswise;  that  is,  multi- 
ply the  first  supposed  number  by  the  last  errour,  and  the 
last  supposed  number  by  the  first  errour. 

If  the  errours  be  alike,  that  is,  both  too  much,  or  both 
too  little,  divide  the'difference  of  their  products  by  the 
Difference  gf  the  errours — the  quotient  will  be  the  an- 
swer. B  it  if  the  errours  be  unlike,  that  is,  one  too  large 

d  the  other  too  small,  divide  the  sum  of  the  products 
[bV  the  sum  of  the  errours. 

EXAMPLES. 

1.  What  number  is  that,  whose  \  part  exceeds  the 
part  by  16? 

Suppose  24;  and  us  £  of  24  is  8,  and  -|  of  it  is  6,  it  is 
(evident  tint  the  third  part  exceeds  the  fourth  part  by  2 
linstead  of  16;  and  therefore  the  errour  is  14  too  small. 
Again,  suppose  48;  and  1  of  48  being  16,  and  £  being 
12,  it  is  manifest  that  the  third  part  exceeds  the  fourth 
by  4,  instead  of  16;  hence  the  errour  is  12  too  small. — 
Then,  the  errours  being  alike,  proceed  thus — 

er. 

1.  supposition  24  \  /\±  too  small. 

/\    er. 

2.  supposition  48  /       \  12  too  small. 

14  672  product.  288  product. 

12  288 

2  dif.  of  er.2)384  difference  of  the  products. 
192  Answer. 

2.  A  son  asking  his  father  how  old  he  was,  received 
this  answer:  Your  age  is  now  j-  of  mine;  but  5  years 
ago,  your  age  was]-  of  mine.     What  arc  their  ages? 

Ans.  20  and  80. 


ARITHMETICAL    PROGRESSION  157 

3.  Two  persons,  A  and  B,  have  each  the  same  income. 
A  stives  ]  of  his;  but  B,  by  spending  50  dollars  per  an- 
nnn  more  than  A,  finds  himself  at  the  end  of  4  years 
3ne  hundred  dollars  in  debt.  What  was  their  income, 
md  what  did  each  spend? 

Ans.  Their  income  was  $125  per  annum  for  each*  A 
spends  $100  and  B  spends  $150  per  annum. 

4r.  What  number,  added  to  the  sixty-second  part  of 
7628,  will  make  the  sum  of  200?  Ans.  77, 

5.  A  man  being  asked  how  many  sheep  he  had  in  his 
irove,  said,  if  I  had  as  many  more,  half  as  many  more, 

us  fourth  as  many  more,  and  12  J,  I  should  have  40. — 
How  many  had  he?  Ans.  10. 

6.  An  officer  had  a  division,  J-  of  which  consisted  of 
nglish  soldiers,  }  of  Irish,  J-  of  Canadians,  and  50  of 
idians.     How  many  were  there  in  the  whole? 

Ans.  600. 

7.  A  servant  being  hired  for  30  days,  agreed  to  re- 
ceive 2s.  6d.  for  every  day  he  laboured,  and  to  forfeit  Is. 
or  every  day  he  played.     At  the  end  of  the  term  his 
>ay  amounted  to  £2.  14s.     How  many  of  the  days  did 

labour?  Ans.  24. 

8.  What  number  is  that,  which  heing  multiplied  by  6, 
the  product  increased  by  adding  18  to  it,  and   the  sum 
divided  by  9,  the  quotient  will  be  20?  Ans.  27, 


ARITHMETICAL  PROGRESSION. 

Arithmetical  Progression  is  a  series  of  numbers  in- 
creasing or  decreasing  by  a  common  difference;  as,  1, 
•2,  3,  4,  5;  1,  3,  5,  7,  9;  5,  4,  3,  2, 1 ;  9,  7,  5,  3,  1,  &c. 
The  numbers  in  a  series  are  called  terms— the  first  and 
last  terms  are  called  extremes,  and  the  common  differ- 
ence is  the  number  by  which  the  terms  in  a  series  differ 
from  each  other,  as  in  2,  5,  8. 11,  &,c. — the  common  dif- 
ference is  3. 

In  any  series  in  Arithmetical  Progression,  the  sum  of 
the  two  extremes  is  equal  to  the  sum  of  any  two  terms, 
equally  distant  from  them,  or  equal  to  double  the  mid- 
dle term  when  there  is  an  uneven  number  o£  terms  in 


ARITHMETICAL    PROGRESSION. 

the  series.  Thus,  in  the  series  2, 4,  6,  8  10,  12,— the 
extremes  are  2  and  12,  equal  to  14,  .md  if  \ou  add  10 
and  4,  or  8  and  6,  the  result  will  Lethe  same;  and  in  the 
series  2,  4,  6,  8, 10,  the  extremes  are  10  and  2,  and 
the  number  of  terms  is  uneven  0  is  the  middleone,  which, 
when  doubled  makes  12,  and  the  extremes  when  added 
together  make  the  same  amount. 

CASE  I. 

The  first  term,  common  difference,  and  number  of  terms, 
being  given,  to  find  the  last  term  and  sum  of  all  the 
terms. 

RULE. 

•Multiply  the  common  difference  by  one  less  than  the 
number  of  terms,  and  to  the  product  add  the  first  term, 
the  sum  will  be  the  last.  Add  the  first  and  last  term* 
together,  multiply  their  sum  by  the  number  of  terms, 
and  half  the  product  will  be  the  sum  of  all  terms. 

EXAMPLES. 

1.  The  first  term  in  a  certain  series  is  3,  the  common 
difference  2,  and  the  number  of  terms  9;  to  find  the  last 
term,  and  the  sum  of  all  the  terms. 

One  less  than  the  number  of  terrne  is  8. 
2  common  difference. 
8  number  of  terms  less  one. 

16  product. 
3-j-  first  term. 

10  last  term. 
3-f-  first  term. 

22 
9X  number  of  terms. 

2)198 

Answer  90  sum  of  all  the  terms. 

2.  A  person  so!d  80  yards  of  cloth  at  3  cents  for  the 
first  yard,  6  for'the  second,  and  "thus  increasing  3  cents 
every  yard ;  what  was  the  whole  amount  ?    Ans.  $97.20 


ARITHMETICAL    PROGRESSION.  1 

3.  How  many  times  docs  a  clock  usually  strike  in 
hours?  Ans.   <8. 

4.  A  man  on  Ji  journey  travelled  20  miles   the   tirst 
»!,j\,^4  the  second,  and  continued  tu  increase  the  nun»- 
.  er  <>f  iniitis  by  every  day  i*>r  10  days      11  »w  tar  diu  he 
tr.ivc  ?  A:is.  3&0  miies. 

.">.  A  firmer  I'uiight  20  cows,  and  gave  2  dollars  f-r 
flu  lirsf,4  f»r  ihe?e^»ud,  and  soon,  giving  in  tlu  same 
j»r  tpoiti'in  ir»m  the  tirst  to  the  last.  Wh.it  did  he  give 
f»r  the  whole?  Ans. 


CASE  II. 

When  the  tiro  extremes  and  Ike  number  of  terms  are  given 
to  fold  t'te  co.nmon  difference. 

RFLR. 

S  i!  tract  the  less  extreme  from  the  greater,  and  divide 
ha  remainder  hy  one  less  dun  the  mimi.er  of  terms — - 
he  <j  io;i;:it  will  I  e  the  common  difference, 

1.  Th>  extremes  i  ei -\£  3  ,.n<l  llv,  a  d  the  number  of 

';',  \vh  it  is  the  comni  >n  difference? 
9  1<>  II 

1  3  13  nutn.of -e  ms. 

'  —  _^  20    fl*1^9' 

12  _J    ' 

Ans.  2  12)10   (in*,  of  extremes 

Common  difference  5    Answer. 

3.  If  the  extremes  le  10  and  ")0,  and  the  numlerof 
'erms  21,  what  is  the  common  difference,  and  the  sum 
»f  the  series?  Ans.  c<  in.  diff.  3,  and  the  sum,  £40. 

4.  A  certain  del  t   can    1  e   p-sid   in  one  yenr,  or  52 
veeks,  by  weekly  payments  in  Arithmetical  Progression, 
he  first  payment  leing  1  dollar,  and  the   lasf  K)3  dol- 
lars.    What  is  the  common  difference  of  the  '(-.us? 

Ans.  $2. 

A  de'^t  is  to  he  discharge!  at  1G  several  payments 
i't  A .'-i^hmetio'.il  Progression;  thetl^t  payme  -t  tole20 
•  1  .rs,  and  the  last  110  dollars.     What  is  the  common 
liffererice?  Ans.  $6. 


60 


GEOMETRICAL    PROGRESSION 


GEOMETRICAL  PROGRESSION. 

Geometrical  Progression  is  the  increase  of  any  series 
>f  numbers  by  a  common  multiplier,  or  the  decrease  of 
anv  series  by  a  common  divisor;  as  3,  0,  12,  24,  48; 
ind  48, 24, 12,  6,  3.  T[\Q  multiplier  or  divisor  by  which 
any  series  is  increased  or  decreased,  is  called  the  ratio. 

CASE  I. 
To  find  the  last  term  and  sum  of  the  scries. 

RULE 

Raise  the  ratio  to  a  power  whose  index  is  one  less 
ban  the  number  of  terms  given  in  the  sum.  Multiply 
the  product  by  the  first  term,  and  the  product  of  that 
nulliplication  will  be  the  last  term:  then  multiply  the 
last  term  by  the  ratio,  subtract  the  first  term  from  the 
jroduct,  and  divide  the  remainder  by  a  nmnl  er  that  ii 
>ne  less  than  the  ratio — the  quotient  will  be  the  sum  ot 
the  series. 

EXAMPLES, 

1.  Bought  12  yards  of  calico,  r,t  2  cents  for  the  firs 
yard,  4  cents  for  the  second,  8  for  -the  third,  &c.:  wha 
was  the  whole  cost? 

NOTE. — The  number  of  terms  12,  and  the  ratio  2. 
1st.  1024  10th.  power, 

term  2  1st.  power. 


4  2d.  power. 

2 

F  3d.  power. 

2 

It!  4th.  power. 
J2 

32  5th.  power. 
32 
64 
06 


2048  llth  power,  or  one  less 
2  2st  term   (than  the  :   m- 


4096 

2  the  ratio. 

8192 


2  .-subtract  the  1st.  term. 

-190   1    is  one  less  than  the 
[ratio. 

24  10th.  power.    $Sl.l)0  Answer. 
2.  B-.MigUt  10  Ibs.  often,  and  paid  2  cents  for  tlio  fir? 
pound,  6  for  the  second,  18  for  the  third,  &c.     Wh;;t  cm 
the  whole  cost?  Ans.  §590.48. 


G  I-:O?,IETI!  1C  AL    PR    G  UESSIOX.  1 1)  i 

3.  The  first  term  in  a  sum  is  1,  'he  who'e  nurn'erof 
4erms  10,  and   the   ratio  2;  wh.it   is  'he  neatest  term, 

;ul  the  sum  of  all  the  tern>?     Ans.  The  greatest  term 
is  512,  and  the  sum  of  the  terms  102'$. 

4.  Whit  debt  may  be  disch  ;rged  in    12  months,  by 
>iviijr  1  dollar  the  first  month,  2  dollars  the   second 
m >nth,  4  the  third  month,   and  so  on,  each  succeeding 
.vivnrmt   being  double  the  last;  and  what  will   be  the 
im  >'int  of  the  last  payment? 

AM*.  The  debt  is  ,§1095,  and  the  last  payment  §2048. 

5.  A   father  whose  daughter  was  married  on  a.  nevv- 
veir's  day,  gave  her  one  cent,  promising  to  triple  it  on 
the  first  day  of  each  month  in  the  year:  what  was  the 
irnoint  of  her  portion?  Ans.  $2657.20. 

0.  Oae  Sossti,  an  Indian,  having  invented  the  g-^me 
of  rhess,  shewed  it  to  his  prince,  who  was  so  delighted 
with  it,  that  he  promised  him  any  reward  he  should  ask; 
;ip  >n  whi°h  Sessa  requested  that  he  might  be  allowed 

me  urn  in  of  wheat  for  the  fir<t  square  on  the  chess  board, 
'2  f  >r  the  second,  4  f  >r  the  third,  and  soon,  doubling1  con- 

inuallv,  to  64,   the  whole   numl  er  of  square?. — Now, 
supposing  a  pint  to  contain   7080  of  these   grains,  arul 

me  quarter  or  8  bushels  to  be  worth  27s.  6d.,  it  is  requi- 
red to  compute  the  value  of  all  the  wheat? 

Ans.  £54481488206. 

7.  VVh-U  sum  would  purchase  a  horse  wi'h  4  shoes,  -UK1 
eight  nails  in  each  shoe,  atone  farlhing  f>r  the  first  nail, 
i  halfpenny  for  the  second,  a  pennv  for  the  third,  &c., 
loubling  to'the  last?  Ans.  £!47£)24.  5s.  3^d. 

8.  A  merchant  sold  15  yards  of  satin,  the  first  yard 
f>r  Is.  the  second  for  2s.  the  third  for  4s.  the  fourth  for 
Ss.  &>o.;  what  was  the  price  of  the  15  yards? 

Ans.  £1038.  7s. 

9.  Bought  30  bushels  of  wheat,  at  2d.  for  the  first 
Hishel,  4d.  for  the  second,  8d.  for  the  third,  &c.;  what 
Iocs  the  whole  amotint  to,  and   what  is    the   price  per 
bushel  on  an  average? 

A         (£Q947848.  10s.  6d.  Amount. 
'    }  £298231.  12s.  4d.  per  bushel. 


2  PERMUTATION. 

PERMUTATION. 

Permutation  is  used  to  show  how  many  ways  things 
may  be  varied  in  place  or  succession. 

RULE. 

Multiply  all  the  terms  of  the  series  continually,  from 
1  to  the  given  number  inclusive;  and  the  last  product 
will  be  the  answer  required. 

EXAMPLES. 

1.  How  many  changes  can  be  made  with  8  letters  of 
the  alphabet? 

1X2X3X4X5X6X7X8=40320  Answer. 

2.  In  how  many  different  positions  can  12  persons 
place  themselves  round  a  tatle? 

1X2X3X4X5X0X7X8X0X10X11X12= 

479001600  Ans. 

3.  How  many  changes  may  be  made  with  the  alpha- 
bet? Ans.  620448401733239489360000. 


SKETCH  OF  MENSURATION, 

OR  PLANES  AND  SOLIDS.* 

Planes,  surfaces,  or  superficies,  are  measured  by  the 
inch,  foot,  yard,  &,c.,  according  to  the  measures  used  by 
different  artists.  A  superficial  foot  is  a  plane  or  surface 
of  one  foot  in  length  or  breadth,  without  reference  to 
thickness.  Solids  are  measured  by  the  solid  inch,  foot, 
yard,  &,c.;  thus,  1728  solid  inches,  that  is  12x12x12 
make  one  cubick  or  solid  f>ot.  Solids  includeall  bodies 
which  have  length,  breadth  and  thickness. 

ARTICLE  I. 

To  measure  a  square  having  equal  sides. 

RULE. 

Multiply  any  one  side  of  the  square  by  itself,  and  the 
product  will  be  the  area,  or  superficial  contents,  in  feet, 
yards,  or  any  other  men  sure,  according  to  the  measure 
used  in  measuring  the  sides. 


*Planes  are  the  same  as  superficies,  or  surfaces. 


SKETCH    OF    MENSURATION 


EXAMPLES. 

Let  A,  B,  C  and  D  represent  a  square,  having  equal 
sides  each  measuring  20  feet  .  Multiply  the  length  of 
one  side  by  itself,  thus  —  20 


20    feet  A 

20  feet 

20 

C 


i 


Ans.  4CO  square  feet. 


B 

20 


D 


20 

2.  How  many  square  rods  are  there  in  a  field  90  rods 
square?  Ans.  8100  square  rods. 

ARTICLE  II. 
To  measure  iJie  plane  or  surface  of  a  parallelogram. 

RULE. 

Multiply  the  length  by  the  breadth — the  product  will 
be  the  superficial  contents. 

EXAMPLE. 

Let  A,  B,  C  and  D  represent  a  parallelogram  whose 
length  is  40  yards,  and  breadth  15  yards. 

40 

Breadth  15    yards  A B 

Length    40  yards  15  Il5 


An?.  600  square  yards.  40 

2.  How  many  square  feet  are  there  in  the  floor  of  a 
room  36  feet  long  and  16  feet  wide?    Ans.  576  square  ft. 

3.  I  engage  to  give  a  plasterer  15  cents  per  square! 
ard,  for  plastering  the  walls  and  ceiling  of  a  room  30| 

eet  long,   15   feet  wide,  *pd  9  feet  high.     How  much 
•vill  his  work  come  to?  Ans.  $21.00. 

Note. — The  contents  of  boards  and  other  articles 
vhich  are  measured  by  feet,  &c.,  may  be  easily  found 
iy  Duodecimal  Fractions. 

ARTICLE    III. 
To  measure  the  plane  or  surface  of  a  triangle. 

RULE. 

Multiply  the  base  by  half  the  perpendicular,  if  it  be  a 
ight  angled  triangle,  and  the  product  will  be  the  area, 
r  superficial  contents;  or  multiply  the  base  and  perpen-i 
icular  together,  and  half  the  product  will  be  the  area. 


1  ,-±  SKETCH    OF    MENSURATION. 

But  if  it  be  an  oblique  angled  triangle,  multiply  half 
the  length  of  the  base  by  a  perpendicular  let  fall  on  the 
base  from  the  angle  opposite  to  it,  and  the  product  will 
be  the  area. 

EXAMPLES. 

1.  Let  C,  II  and  G  represent  a  right  angled  triangle, 
having  the  right  angle  at  G ;  the  base  C  G  being  40  feet, 
and  the  perpendicular  II  G,  28  feet. 

No.  1. 

14    feet,  or  half  the  perpendicular.  .  H 

40  feet,  or  the  base,  ^'^     •£ 

5oO  feet — the  area.  vS 

!  rise. 


2.  Let  B,  C  and  D  represent  an  oblique  angled  tri- 
ingle;  the  length  of  the  base  B  D  being  60  feet,  and 
the  perpendicular  C  E,  28  feet.  No.  2. 

28    the  perpendicular  C 

40  half  the  base. 

1120  Answer 


SO  R 

3.  How  many  square  rods  are  there  in  a  triangular 
field,  one  of  the  corners  of  which  is  a  right  angle,  and 
one  of  the  shorter  sides  of  which  is  3d  rods,  and  the  oth*r 
24  rods?  Ans  432  square  rods. 

NOTE. — Right  angled  triangles  arc  tuch  as  hare  one 
angle  like  the  corner  of  a  xquarc,  and  which  is  called  the 
rig/it  angle,  containing  90  degrees;  as  the  angle  G  in 
the  triangle,  No.  1 . — Oblique  angled  triangles  arc  sw-h 
as  have  each  of  the  angles,  either  more  or  less  than  IK) 
degrees*  as  in  the  triangle,  No.  2. 

ARTICLE  IV. 

To  me  as  re  a  circle. 

Note. — Circles   are   round    figures,    1  ounderl    every 

where  by  a  circular  line,  called  the*  periphery,  and  aUo 

the  circumference.     A  line  passing  through  the  cei.fpa 


SKETCH    OF    MENSURATION. 


105 


is  called  the  diameter.     Half  the  length  of  the  diameter 
is  called  the  radius. 

The  diameter  may  be  found  by  the  circumference, 
thus — As  '£&  is  to  7  so  is  the  circumference  to  the  diame- 
ter; and  in  like  manner  may  the  circumference  be  found 
by  the  diameter;  for,  as  7  is  to  22,,  so  is  the  diameter  to 
ihe  circumference. 

1.  What  is  the  diameter  of  a  wheel,  or  circle,  whose 
circumference  is  16  feel?  Ans.  5  feet,  nearly. 

2.  What  is  the  circumference  of  a  circle,  whose  diam- 
eter is  2i)  feet?  Ans.  63  feet  nearh  . 

i>.  if  the  distance  through  the  earth  be  b>000  miles, 
how  many  miles  around  it?  Ans.  25143  miles  nearly. 

ARTICLE  V. 

To  find  tlie  superficial  contents,  or  area,  of  a  c  ircle. 

JiULE. 

Multiply  half  the  circumference  by  half  the  diameter, 
rid  the  product  will  I  e  the  answer.  Or,  multiply  the 
qii'ire  of  the  .'iameter  by  .7&54;  or  multiply  the  square 

of  the   circumference  by  .07958,  and  in  either  case  the 

product  will  be  the  answer. 

EXAMPLE. 

How  many  square  feet  are  contained  in  a  circle  whose 
circumference  is  44  feet,  and  whose  diameter  is  14  feet? 
22  half  the  circumference. 
7  half  the  diameter. 

154  square  feet.     Answer. 

The  same  may  be  done  by  multiplying  the  diameter 
ind  circumference  together,  and  dividing  the  product 
•y  4,  thus,  44x14=616-7-4=:  154.  Answer. 

2.  H  >w  many  square 'feet  are  there  in  the  area  of  a 
ircle  whose  circumference  is  16  feet? 

Ans.  20  square  feet. 

3.  Hv>w  many  sq-iare   feet,  are  there  in  the  area  of  a 
ircle  whose  diameter  is  20  feat?      Ans.  315  sqiriro  ft. 

ARTICLE  VI. 
To  measure  the  surface  of  a  globe  or  sphere. 

RULK. 

Multiply  the  circumference  by  the  diameter,  the  juo- 
duct  will  be  the  surface,  or  arc. 


16t>  SKETCH    OF    MENSURATION. 

EXAMPLES. 

1.  What  are  the  superficial  contents  of  a  globe  whose 
circumference  is  220  feet,   and    v.  h;.-se  diameter   is  "70 
feel?  220  X  H;  =  1,3400  square  feet.     Answer. 

2.  How  n>  ny  Mjuare  nii.es  ;  re  conuiLied  on  the  sur- 
face of  the  vshole  earth,  or  globe,  vvhi<  h  we  iiihal  i-? 

The  circumference  of  the  earth  is  estimated  to  le 
25020  miles,  and  the  diameter,  7V.64,  nearly. 
Then,  25020x79u^  =  191259280  square  miles.  Ans. 

ARTICLE  VII. 

To  find  the  &olid  coidtnts  of  a  cube* 
RULK. 

Multiply   the  length  of  one  side  1  y  itself,  and  muUi 
ply  the  product  1  y  ihe  sjane  length,  th.it  i«,  by  the  same 
multiplier-  the  last  product  will  Le  ihe  solid  contents  of 
the  cube. 

EXAMPLES. 

1.  How  many  solid  feet  are  contained  in  a  cul  e,  cr 
solid  block  oft>  equal  sidos,  each  side   Icing  3   feet  in 
length,  and  3  in  breadth? 

3X3X;>=27  solid  or  cu1  icfcfeet.     Ans. 

When  the  contents  are  required  of  right  angled  solids  J 

[whose  length,  breadth,  etc.,  jsre  not  equal ;  multiply  th<i 

liength  l»y  the  1  readth,  and  that  product  by  ihe  thickness, 

I; he  pn  dir-t  will  be  the  «  nswer. 

2.  Req-i'r  d  the  con  ents  of  a    1  ;ad  of  wood,  whoi-e 
length  is  8  feet,  breadth  4  foet,  a1  d  height  or  thickness 
[  feet.  8x1  X  1=  128  solid  f ;e%  or  1  cor'!.     Ans. 

3.  Require  1  the  contents  of  a  j-torie  L*>J  Let  in  leng:b, 
'A  in  bre  ic'th,  and  1  fx»t  in  Lhickne.-s. 

16.5X^.5X1=24^75  solid  feet,  or  1  perch.     Am-. 
Note. — S  )lids  whose  climeriFions  Jire  in  feet  or  inches, 
ire  more  easily  measured  by  Duodecimals. 

AUTICLE  VIII. 
To  find  the  content*  of  a  privm. 
A  prism  is  an  angular  figure,  generally  of  three  eqi^l 
>i'Jes,  whose  ends  are  in  the  f.»rm  of  trinr-glcs.     It   rt-- 
jsembles  a  file  of  three  sijes,  whose  whole  length  is  <;t 
equal  bigness. 

*A   cubo   is  a  solU  boily  of  equal  sides,  each  of  which  is  an 
cxactsqunrc. 


KKN31IKATION. 


157 


RULE. 

Find  the  area  or  superficial  contents  of  one  end  as  of 
diw  other  triangle,  then  multiply  the  area  by  the  length 
jf  the  prism,  and"  the  product  will  Le  the  soadity. 

EXAMPLE. 

What  are  the  solid  contents  of  a  prism,  the  sides  of 
he  triangles  of  which  measure  13  inches,  the  perpendic- 
ilar  extending  from  one  of  its  angles  to  its  opposite  side, 
[2  inches,  and  its  length  18  inches? 
13 X  l^=li>t)-r~2—  'aX  !&•=  140-1  cubick  inches.  Ans. 

To  find  the  coiittfittj  of  a  cylinder. 
A  cylinder  is  a  lo.ig  ro.uul  body,  ail  its  length  being 
of  equal  bigness,  like  a  round  ruler. 

RULE. 

Find  the  area  of  one  end,  t>v  the  rule  f  >r  finding  the 
area  of  a  circle,  then  multiply  it  by  the  length,  and  the 
product  will  be  the  answer. 

EXAMPLE. 

What  is  the  solidity  of  a  cylinder,  the  area  of  one  end 
>f  which  contains  ^.40  square  feet  and  its  length  being 
12.5  feet?  2.40 X  1^.-~>=30  solid  feet.  Ans. 

ARTICLE  X. 
To  find  the  solid  content*  of  a  round  &tick  of  timber,  which 

wr  of  a  true  taper  from  the  larger  to  the  smaller  end. 

RULE. 

Find  the  area  of  both  en-Is;  add  the  two  areas  together 
md  reserve  the  sum;  multiply  the  area  of  the  larger  end 
;>y  the  area  of  the  smaller  end,  extract  the  square  root 
•  »f  the  product,  add  the  root  to  the  reserved  sum,  then 
multiply  this  sum  by  one  third  the  length  of  the  stick, 
md  the  product  will  be  the  solidity. 

NOTE. — As  this  method  requires  considerable  labour,  the 
following  has  been  preferred  for  common  use,  though  not 
julle  sj  accurate. 

RULE. 

Girt  the  stick  near  the  middle,  hut  a  little  nearer  to 
Uie  larger  than  to  the  smaller  end ;  this  will  give  the  cir 
oumference  at  that  place.  Fi  id  the  diameter  by  the  cir- 
cumference; m  i.tiply  the  circumference  and  the  diam- 
eter together;  then  multiply  one  fourth  of  the  product  by 
the  length  and  the  a  nsiver  will  i.enearlv  the  solid  contents. 


&  PRACTICAL    QUESTIONS. 

EXAMPLES. 

What  is  the  solidity  of  a  round  stick  of  timber  that 
is  10  feet  long,  and  its  circumference  near  the  middle 
is  2.61  feet? 

As  22   :  7   :   :  2.61    :  .83  diameter, 
cir.  diam.  length,  feet. 

2.61  X.83=2.1663-r4=.5415x  10=5.4150. 

Ans.  5.4150  solid  feet. 
ARTICLE  XI. 
To  find  the  solid  contents  of  a  globe, 

RULE. 

Multiply  the  cube  of  the  diameter  by  .5236,  the  pro- 
duct will  be  the  solid  contents.  Or,  multiply  the  super- 
ficial contents,  or  surface,  by  one  sixth  part  of  (he  sur- 
face. Or,  multiply  the  cube  of  the  diameter  by  11,  and 
divide  the  product  by  21 — in  either  case  the  product 
will  be  the  solidity. 

EXAMPLES. 

1.  What  are  the  solid  contents  of  a  globe  whose  di- 
ameter is  14  inches? 

14X14XM=2744X.5236=1436.7584 

cubick  inches.     Ans 

2.  How  many  solid  miles  are  contained  in  the  earth, 
or  globe,  which  we  inhabit? 

Suppose  the  diameter  to  be  7954  miles;  then, 7954 X 
7954x7^4=503218686664  the  cube  of  the  earth's 
axis,  or  diameter;  then, 

503218686664  X  .5236=263485304337 

cubick  miles.  Ans. 
Note. — The  solidity  of  a  globe  may  be  found  by  the 
circumference,  thus — Multiply  *he  cube  of  the  circum- 
ferance  by  .016887 — the  product  will  be  the  contents. 


PRACTICAL  QUESTIONS. 

1.  A  cannon  ball  goes  about  1500  feet  in  a  second  of 
time.  Moving  at  that  rate,  what  time  would  it  take  in 
going  from  the  earth  to  the  sun;  admitting  the  distance 
to  be  100  millions  of  miles,  and  the  year  to  contain 
365  days,  6  hours?  .  Ans.  lOy^yW  years. 


PRACTICAL    QUESTIONS.  109 

2.  A  young  man  spent  £  of  his  fortune  in  8  months,  ^ 
of  the  remainder  in  12  months  more,  after  which  he  had 
£410.  left.     What  was  the  amount  of  his  tVtune? 

Ans.  £956.  13s.  4d. 

3,  What  number  is  that,  from  which  if  you  take  |  of 
|,  and  to  the  remainder  add  T7?  of  J^,  the  sum  will  be 
10?  Ans.  lO^u 

4.  What  part  of  3,  is  a  third  part  of  2?         Ans".  f . 

5,  If  20  men  can  perform  a  piece  of  work  in  12  days, 
how  many  will  accomplish  another  thrice  as  large,   in 
:me  fifth  of  the  time?  Ans.  300. 

(J.  A  person  making  his  will,  gave  to  one  child  ij  of 
his  estate,  and  the  rest  to  another.  When  these  lega- 
cies were  paid,  the  one  proved  to  be  £600  more  than 
the  other.  What  was  the  worth  of  the  whole  estate? 

Ans.  £2000. 

7.  The  clocks  of  Italy  go  on  to  24  hours ;  how  many 
strokes  do  they  strike  in  one  complete  revolution  of  the 
index?  Ans.  300. 

8.  What  quantity  of  water  must  be  added  to  a  pipe  of 
wine,  valued  at  £33,  to  bring  the  first  cost  to  4s.  6d. 
per  gallon?  Ans.  20|  gallons. 

9.  A  younger  brother  received  £6300,  which  was  7 
f  his  elder  brother's  portion.     What  was  the  whole  es- 
tate? Ans.  £14400. 

10.  What  number  is  that  which  being  divided  by  2,  or 
3,  4,  5,  or  6,  will  leave  1  remainder,  but  which  if  divi- 
ded by  7  will  leave  no  remainder?  Ans.  721. 

11.  What  is  the  least  number  that  can  be  divided  by 
the  nine  digits  without  a  remainder?          Ans.  2520. 

12.  How  many  bushels  of  wheat,  at  $1.12  per  bushel, 
can  I  have  for  $81.76?  Ans.  73. 

13.  What  will  27  cwt.  of  iron  come  to,  at  $4.56  per 
cwt.?  Ans.  $123.12. 

14.  When  a  man^s  yearly  income  is  949  dollars,  how 
much  is  it  per  day?  Ans.  $2.60. 

15.  My  factor  sends  me  word  he  has  bought  goods  to 
the  v.; hie  of  £500.  13s.  6d.  upon  my  account;  what  will 
his  commission  come  to  at 3 l  per  cent.? 

Ans.  £17. 10s.  5^d. 


170  PRACTICAL   QUESTIONS. 

16.  How  mnny  yards  of  cloth,  at  17s.  6d  per  yard,  can 
I  have  for  13  cwt.  2  qrs.  of  woo!,  at  14d.  per  1!  .? 

Ans.   1 00  yards,  8 £  qrs. 

17.  There  is  a  cellar  dug  (hat  is  12  feet  every  way,  in 
length,  I  m«dth,a   d  depth;  how  many  solid  feet  of  earth 
were  taken  out  of  i  ?  Ans.   172R. 

18.  If  2.  of  ari  ounce  cost  J  of  a  shilling,  what  will  £ 
of  a  11).  cost?  Ans.  17s.  6d. 

19.  If  £  of  a  gallon  cost  £  of  a  £.  what  will  £  of  a  tun 
cost?  Acs. '£105. 

20.  Iff  of  a  ship  be  worth  £3740,  whnt  is   the  worth 
of  the  «  hole?  Ans.  £.1973.  6s.  8d. 

21.  What  is  the  commission  on  $2176.50,  at  ££  per 
cent?  Ans.  $54.41J. 

22.  In  a  certain  orchard  £  <  f  th'^  trees  1  e  tr  nj  pb>s, 
pears,  J-  plums,  60  of  tl  e  '.i  pearlvs,  arid  40  cherries; 
how  many  trees  are  in  the  orchard?  Ans.   12(0. 

23.  If  A  travel  by  m  *il  at  the  rate  of  8  miles  an  h'  ur, 
and  when  he   is  50  miles  on  his  way,  B  start  from  the 

wo  place  that  A  di;?,  and  travel  on  horseback  the  s;  me 
road  at  10  miles  an  hour,  how  long  and  how  far  will  B 
travel  to  come  up  with  A? 

Ans.  25  hour?,  and  250  miles. 

24.  Bright  a  quantity  r.f  cloth  f  r  750  dollar?,  ^  of 
vvhi  vh  1  found  to  be  inferior  which  I  Ind  tosell  at  1  dol- 
lar 25  cents  per  yard,  and  by   this  I  lost   100   dollars: 
what  must  I  sell  the  rest  at  per  yard  that  I  shall   lose 
nothing  by  the  whole?  Ans.  $3.15^f.. 

25.  If  the  ear  h  goes  rrund  the  sun  once  in  365  days, 
5  hours,  48  minutes,  49  seconds,  and  its  distance  from 
the  sun  95000000  miles,  what  must  be  the  distance  of 
the  planet  Mercury  from  the  Sun,  admitting  the  time  of 
its  revolution  round  the  Sun  to  be  87  days,  23  hours,  15 
ninutes, 40  seconds? 

Note. — The  planets  describe  equal  areas  in  equal 
fimes  therefore,  as  the  square  of  the  time  of  the  revo- 
lution of  one  planet,  round  the  Sun,  is  to  the  squ  re  of 
he  time  of  the  revolution  of  any  other  planet,  so  is  the 
•u!  e  of  the  distance  of  one  planet  from  the  S-in,  to  the 
^\}  o  "f  th«  .'•'ist -nee  of  nnv  other  from  the  S  'n. 


171 
A  SHORT  SYSTEM 

OP 

BOOK-KEEPING* 

FOR 

FARMERS  AND  MECHANICS. 


is  the  method  of  recording  bus  ness  transactions. 
It  is  of  two  kinds — single  and  double  entry ;  but  we  shall  on.y 
lotice  the  former. 

Single  entry  is  the  simplest  form  of  Book-Keeping  and  isem- 
)lojed  by  retailers,  mechanics,  farmers,  occ.  It  requires  a  Day- 
JOJK,  Leger,  and  where  money  is  frequently  received  and  pai,. 
out,  a  Cash-Book. 

A  few  examples  only  are  here  given,  barely  sufficient  to  givi 
the  learner  a  view  of  the  manner  of  Keeping  books;  it  being  in- 
tended that  the  pupil  should  be  required  to  compose  similar  one*, 
ind  insert  them  in  a  book  adapted  to  tnis  purpose. 

The  Daj-B  > jk  contains  entries  of  tne  several  articles  in  the 
successive  order  of  iheir  dates.  Each  person  must  be  mule  UY. 
for  what  he  receives,  and  (Jr.  by  wnat  is  received  of  him  o.. 
account. 

Every  month,  or  oftener,  the  Day-Book  should  be  copied  01 
posted  into  the  -.eger,  as  hereafter  diiectcd.  The  crjsse v>n  tin 
ieft  hand  column,  show  that  the  charge  or  credit,  agam-t  wmc,. 
they  st  ind,  is  ported,  and  the  ligures  show  tlie  page  of  the  i^egei 
where  the  account  is  posted.  Some  use  the  tig-are*  only  as  poo* 

Tne  Leger  is  the  grand  book  of  account?,  in  which  every  pcr- 
f^s  account  is  collected  from  different  j>arts  oi' the  Day  Book, 
and  inserted  in  one  place;  the  Dr.  and  v.-r.  fronting  each  «>thei 
ou  opposite  pages  or  on  opposite  sides  of  the  same  page,  whici 
shows  the  w'hole  state  of  the  account  at  once. 

i^ost  or  transfer  the  entries  from  the  Day  Book  to  the   Leger. 
thus:  Open  an  account  in  the  Leger  for  the  first  person  vvt 
stands  JJr.  or  CV.  iu  the  Day  Book,  i.  e.  write  his  name  with  D 
m  the  left  hand  page  of  the  folio,  aud  Cr.vii  the  richt. 


172  DAYBOOK. 


January  18th.  1831. 


IX 


IX 


Peter  Simpson  of  Cin*       Dr. 

To  15  yards  of  fine  Broad-cloth,  a  5.00      $75.00 
"  24  do.  superfine  do.  a  7.75    .     .     .       186.00 

• 

R.  Fulton^  of  Newport,      Dn 

To  1  gall.  Molasses $    50 

"  6  ibs.  Coffee,  a  37*  cts.            .     .     .        2.25 
"  20  Jbs.  Sugar,  a  10  cts 2.00 


I          j  -  -  19 


X 


IX 


IX 


John  CatJiell,  Carpenter,    Dr. 

To  16  vards  Calico,  a  I2i  cts.     .    •    .      $2.00 
"  10""      Muslin,  a  15  cts.  .     ,    .    2.50 

•*    1  Vest  pattern,      ........  75 

*'     1  pair  Gloves,  .........  62 

14  25  Ibs.  Nails,  a  8  cts  .......     2.00 


Charles  H.  Glover,          Dr. 

To  25  Reams  post  paper,  a  $3.00,    .     .     $75.00 
18     "    foolsca  >  do.  a    3.25,    .    .       56.50 


George  Whipple,Jr.         Dr. 

To  300  Ibs.  Pork,  a  5  cts $15.00 

50  bu.  Corn,  a  20  cts 10.00 

_20 


William  Jones,  Dr. 

To  35  Ibs.  Iron,  a  7  cts $J.45 

4i  Cash  paid  his  order  to  John  Bnker,  1.35 


James  L.  Rowan,  Dr. 

To  50  bis.  Flour,  a  $3.50,    ....      $175.00 

"  25  bu.  Potato*,  a  30  cts 7.50 

"    4  bis.  Cider,  a  1.50, 6,00 

*•  75  Ibs.  Beef,  a  4  cts 3.00 


Peter  Simpson,  Cr. 

;By  30  cords  Wood,  a  $2.25,    ....  $67.50 

"  90  bu.  Oats,  a  12i  cts 11.25 

"    5  tons  Hay,  a  15.00, 75.00 


1*1 

DAY  BOOK. 

I 

7^ 

i 

January  "2  1  at.  le#  1. 

ix 

Burchel  J.  Barney            Dr. 

Po  6  galls.  Port  u  me.  «  $J.DJ,     .     .     .     $15.  M) 
"   -JlGibs.  Sugar,  a  vJcts.         ....       1:J.3.'I; 
"  1  ib   Tea,     1.  J51 

$ 

C. 

35! 

60 

ix 

Cumtims  C.  Williams,      Dr. 

To  17  >  bis.  Whisky  5;9  J  gulls,  a  •>{)  cts.  $1J38.00 
44  Paitl  liis  onlcvin  j'avor  of  'J^honias 
and  miite,  ibr  salt,     234.  ')0 

! 
| 

1 

C/. 

By  750  Ib?.  Feathers,  a  25  cts.     .     .     .     $187.  )0 
»'i  Ca«h,                                                          5  IJ  0 

]  ^72 

,)0 

—  1 

6S7 

50  I 

ix 

John  Pkares,                      Cr. 

•K  sundries  Tor  which  1  .i^Jivc  rny  note  at  60  <lnyc, 

184 

50 

2X 

Peter  SitnpWH,                  Dr. 

'o  order  ou     ()l)crt  F*ult<Hi«          •          .     . 

1 

37 

2X 

ttowrt  l^uitoff,                   Cr.  i 

[5v   i"«'t(  r  -^iinnson's  order  on  him, 

1 

37 

bx 

William  JoneS)                  Dr. 

Po  14  Ib-.  Veal,  «  4  cts  g     .5* 

i 

"  ^00  Ibs.  Flour,    2.5 

Or. 

By  his  bill  of  blacksmith  work,    

3 

f 

.)() 

03 

9 

2X 

John  Cathcll,                     Cr. 

By  4  days  carpenter  work,  a  $1.25,    .     ,     $5.0 
"  ^00  feet  Poplar  boards,  a  31  d  cts.     .     .     1.7" 

i 

75 

8X 

George  Whipple,  Jr.          Cr. 

!>y  Cash  on  account,     .          

30 

Oi;  ! 

2X 

Claudius  C.  Williams,      Cr. 

By  10  bis   Mackerel,  No.  1,  a  $12.00,       $1'20.0:> 
u  1-20  galls.  Cog.  Brandy,  a    1.50,.     .     180.00 

1 
! 

£00 

oo  !: 

2X 

John  Phares,                     Dr. 

To  cash  on  account  of  my  note  at  60  day*, 

174 

DAY  BOOK. 

Pi) 

January  24th,  1831. 

IX 

1 

Robert  Fulton,                  Dr. 

Fo  1  piece  Broadcloth  containing  25  yards, 
a  $6  per  yd.  ;  90  days'  credit,     .... 

*    < 
150  ( 

)0    1 

ix, 

Peter  Simpson,                  Cr, 

3y  Cash  in  full,     

08 

32    1 

2X, 

Win.  Jones,  Blacksmith,    Dr. 

Fo  217  Ibs  Iron,  a  S  cts  . 

17 

36  11 

Cr. 

i 

25  II 

O1^ 

IX 

Charles  H.  Grover,           Dr. 

o  rent  of  my  house  3  months,  a  $6  per  month, 
Cr. 

18 

0    1 

"  Cash,      100.00 

112 

'  II 

IX 

George  Whipple,Jr.        Dr. 

'o  1  Keg  nails,  weighing  215  Ibs.  a  8  «ts.    .    . 
Ofi 

17 

20    1 

2X 

James  L.  Rowan,               Cr. 

By  use  of  his  horse  20  days,  A  50  eta.    .    $10.00 
"  1500  Ibs.  Rags,  a  4  cts  60.00 

"     160  bu.  Salt,  a  50  cts  80.00 

150 

00 

2X 

Burchel  J.  Barney,          Dr. 

To  1  Bag  of  coffee  weighing  216  Ibs.  a  16  cts. 

34 

56    1 

3X 

C.  Smith,  of  Lexington,     Dr. 

To  15  Ibs.  Wool,  a  25  cts  $37 

"  20  Ibs.  Flax,  a  9  cts  18 

II 

55    I 

*X 

A.  Dunn,  of  Columbus,      Dr. 

To  800  ft.  Pine  boards,  a  $1.25,     .     .      $10.0 
"     84  ft.  Scantling,  a  3  ct?  2.5 

IX 

Robert  Fulton,                 Cr. 

By  1200  Ibs.  Pork,  a  4  cts  

1 
4 

52 
(\(\  i] 

IX 

William  Jones,                  Cr. 

,By  Cash,    

1 

30 

I43 

DAY  BOOK. 

175  |j 

January  21th.  1831. 

±x 

John  Cathell, 

o  350  lbc  Nails  a  8  cts    .     .     .    . 

Dr.    $ 

.    *28.00 

a  U 

1  17  Door  locks,  a  1.^5,    .     .     .     • 

.    *dl.-^5 

41 

)  25    1 

IX 

Cftarles  H.  Graver, 

y  Cash  on  account,     

Cr. 

.  .       i 

0  37 

2X 

Claudius  C.  Williams, 

To  856  Ibs.  Tobacco,  a  5  cts.     .     .     . 

Dr. 

...       4 

280    I 

IX 

George  Whipple,  Jr. 

By  1700  Ibs.  Cheese,  a  7  cts.     .     .     . 

Cr. 

.   .   .      u 

900 

2X 

James  L.  Rowan, 

To  24  yds.  Linen  shirting,  a  87  £  cts. 

Dr. 

>1  00    I 

IX 

John  Cathell, 

By  Cash  in  full,                   .... 

Cr. 

=iO  37  A  1 

°8 

)U  0<  £    1 

3X 

Calvin  Smith, 

By  41  Ibs.  Coffee,  a  18  cts.     .     .     . 

Cr. 

.     .     $7.38 

"  36  Ibs.  Sugar,  a  12i  cts.    .    .    . 

.     .      4.50 

11  88    1 

IX 

George  Whipple,Jr. 

By  12  days  labour  of  self,  a  62*  cts. 
u    7  days  do.  of  his  horse,  a  25  cts. 

Cr. 

::  ^ 

925    1 

2X 

Burchel  J.  Barney 

By  cash  in  full,    

Cr. 

70  25    I 

2X 

Andrew  Dunn, 

To  mending  Wagon,     ..... 

Dr. 

«5.75 

"  Timber  and  materials  for  do.     . 

.     .     1.25 

700    I 

2X 

James  L.  Rowan, 

By  Cash  in  full,     

Cr. 

62  50  || 

1 

2X 

John  Phares, 

To  730  Ibs.  Sea  Island  salt,  a  15  cts. 
of  my  note  

Dr. 

in  full 

1HQ  en   II 

3X 

Calvin  Smith, 

To  75  yds.  Domestic  cloth,  a  50  cts. 

Dr. 

3    50  |! 

170 
FOUM  OF  A  LEGEI?.                 [1 

Dr.               Peter  Simpson,              Cr. 

1  1831.1 
;Jan.l8l 

"    222 

11 

To  Sundri.-.-, 
«  Order  on  R.  F. 

.  $  \  < 

I'l 

•'    1831.- 
-  Jan.  20 

?    u    24 

i>y  sundries,    1£ 
3    "  Cash,         H 

!'•>( 

r 

»    c. 
375 
862 

np 

Dr.               Robert  Fulton  9-             Cr 

Ian.  181 

«     243 

To  Sundries,    i     475  ,1 
"  Broadcloth  J150  00 

[  154  75 
"  Balance,  .  UO^, 

1831. 
an.  22  2  F^y  Order  P.  S. 
tk    2tiS   *'*•  Pork,     .     .      ' 
"  Balance,           l( 

18  )0 
)5    8 

Dr.                 John  CatJiel'l,                Cr. 

1831.  I 
Jan.  1911 

"    274 

To  Sundries, 

[   "      do. 

57  l2£ 

Jan.  22  . 
"     27  4 

By  Sundries, 
"  Cash,            5 

5 

i- 

^   c. 
G75 
J37i 

7  12* 

Dr.          Charles  H.  Grocer,          Cr. 

Jan.  19 
"     25 

To  Paper, 
3  «  Rent,    . 

"  Balance, 

133  5*1 

15  :  -k 

)Q    ^  1 

.     1831. 
)  Jan.  25 

,    "     27 

> 

3  Bv  Sundries,     1 
1    <;  Cash, 
"  Balance, 

i 

i 

10;37 

28|51 

51  ">  •.) 

Dr.          George   Wkippic,Jr.          Cr. 

1831. 
Jan.  19 
«     25 

1 

1  To  Sun  hies 

3,  «  Xai  *, 
"  Balance 

i 

i*7  I 
10605 

!4S  25 

}  1831. 
Jan.  23 

l-     27  -J 
"     28^1 

i 

\  By  Cash, 
"  Cheese,          I 
"  Sundries, 

1 
"  Balance,      ;1 

20  00 
1900 

9:25 

4a  25 
06  05 

|2]                  FORM  OF  A  LEGFR.               l77 

Dr.               William  Jones, 

1831. 

$  c. 

1831. 

c 

f.-\ 

Jan.  20 

"     22! 

1  TQ  Sundries, 
2  «    do. 

370 
306 

Jan.  22  2 

"     243 

By  Bill  work, 
"    do. 

1 

57 
25 

•*     24 

3  "  Iron, 

1736 

«     263 

44  Cash, 

17 

30 

2412 

in 

24 

12 

Dr.            James  L.  Rowan,            Cr. 

l&U. 

$    < 

.    183  i. 

$ 

c. 

Jan.  20 

i  To  Sundries, 

1915 

0  Jan.  26 

3  By  Sundries, 

JO 

«    37 

i  "  Linen, 

210 

0    "     28 

4   "  Cash, 

62 

50 

2125 

0 

212 

50 

Dr.           Bureliel  J.  Barney,          Cr. 

l63l. 

$  c 

.     1831. 

.f 

e. 

Jan.  21 

2  To  Sundries, 

356 

3  Jan.  28 

4  By  Cash, 

70 

25 

"     26 

3  "  Coffee, 

345 

6 

70  2 

5 

... 

Dr.        Claudius  C.  Williams,         Cr. 

1831. 

$     c 

1831. 

$ 

c. 

Jan.  21 
27 

2  To  Sundries,  i 
4   «  Tobacco, 

272  OC 

42  8f 

Jan.  21 
"     23 

2  By  Sundries, 

300 

5\ 
00 

u  Balance, 

327 

30 

I 

31481 

1 

314 

80 

"  Balance, 

327  3( 

Dr.                John  Fhares,                Cr. 

1831. 

$    c- 

II  1831. 

$ 

c. 

Jan.  2? 

2  To  Cash, 

7500 

Jan.  2^  • 

By  Sundries, 

184 

50 

"     28 

1   «  Cotton, 

10950 

] 

18450 

. 

Dr.               Andrew  Dunn,               (Jr. 

1831. 
Jan.  26J; 

*  To  Sundries, 
a  Balance, 

I 

f   1831. 
Jan.  28J4 

By  Sundries, 
"  Balance, 

$ 
7 

e. 

f)0 

i 

12 

ii 

178                        CASii-LOOIi.                         [3 

Dr.               Ctdmn  Smith,                Cr. 

1831.    |                              $     c.  ' 
Jan.  :2G'3To  Sundries,     5    55 
44     28  J4   "  Cloth,         37    50 

43    05 
1     "  Balanco,    31    15 

1631.                                $     c 
Jan.  •&  4  By  Sundries,    1  1    $8 
"  Balance,     31    15 

43   05 

INDEX  TO  LEGER. 

R                  Pace: 
Barney,  Burchel  J.     .     .    2 

P                F*gt. 

Vhare?,  John  ....    2 

C 

Cathell,  John    ....    1 

Rowan,  James  L.     .     .     2 

Dunn,  Andrew      ...     2 

s 

Simpson,  Peter     ...     1 
Smith,  C  akin       .     .          3 

F 

Fulton,  Robert     ...     1 

1^ 

Grover,  Charles  H.     .     .    1 

w 

Whipple,  Georee    .     .     1 
Williams,  Claudius  C.    .   2 

j 

•91 

BOOK. 

its  and  receiptt  of  cash. 
r.  to  cash  on  hand  and  what  is 
paid  out. 
^ek,  as  may  best  snit  the  nature 
1  is  counted,  and  entered  on  the 

ake  the  sum  of  the  Dr.  equal  to 
en  struck,  and  the  cash  on  hand 

i 

J 

.lories,  \Villiuin     .     .     .     2 

•pi 
CASH- 

This  book  record*  the  paymei 
It  is  kept  by  makinsr  cash  D 
received,  and  Cr.  by  whatever  is 
At  the  end  of  every  day  or  w< 
of  the  business,  the  cash  on  han 
Cr.  side. 
If  there  i=  no  error,  this  will  m 
that  of  the  Cr.     A  balance  is  th 
carried  again  upon  the  Dr.  side. 

Dr. 


OF  A  CASH-liOUK. 

CASH. 


831: 

Jan.   1  To  cash  on  hand 
2  «  J.Bali 
44  44  E.  Jennison 
S  44  D.  Roe  paid) 

**  44  H.  Austin  on) 
his  note          j 

4  "  8.  Ball  on  ace 

5  44  A.  Higby 

6  44  Sales  of  Aer-) 
\     chandise         ) 


I  3 

t!     * 

is 

3H 


idol., 


Jan.  1  By  ain't.  |  aid 

repairs 


•epj 

41  Paid  note  to, 
A.  Y\  estou  ' 
44  Paid  work  nu 
44  Family  ex- 
pense's 

44  Merchandise) 
bought  at      £ 
auction          j 
(.'ash  on  hand 


8  Cash  on  hand 

^••••••••MiMMB    JMB^MB 

.Form  of  a  Bill  of  parcel*  jrom  the  preceding  Work. 
Mr.  John  Catliell 

To  Solomon  Thrifty,     Dr. 

1831.  I  ~ 

Jan.  16.     To16y(KCalicf>,  al2J  cts.     .     .     .      $3.00 

u      a        "10yds.  Mnslin,  a  15  cts 1.50  | 

«      «        a  i  vest  pattern, 75  I 

44      "        44  1  pair  (ilove?, ,         6^4' 

«      44        44  25  Ibs.  Nails  a  8  cts 2.00 

44    27       44  350  Ibs.  Nails  a  8  cts 28.00 

44     "       "  17  Door  locks  a  $1.25  .     ....      21.25 

49 

Cr.  56 124 

44     5      By  4  days  Carpenter  worker  $1.25  .     .    .$5.00 
44      "       "  200  ft.  Poplar  boards  a  874  cts.     .     .     1.75 

44    27      "  Cash  in  full, 43.374 

Errors  excepttd.  !56  124 

CINCINNATI,  Tan.  27th,  1831.  SOLOMON  THRIFTY. 

2rf  Form. 
Claudius  C.  Williams, 

To  Solomon  Thrifty,     Dr. 

1831.  $    c. 

Jan.  21.    To  172  bis.  Whiskey,  5190  galls,  a  20 

cts $1038.00 

44      44        44  Paid  his  order  favour  of  Thomas 

White,  for  Salt 234.00 

137? 00 

«    27      «  856  Ibs.  Tobacco,  a  5  cts 4780 


CONTENTS 

PAGE, 

Numeration            -  7 
Simple  Addition            *                         .9 

Simple  Subsiraction            -  13 

Simple  Multiplication  17 

Simple  Divison  22 

Federal  Money  33 

Table  of  Money,  Weights,  Measures  &c.  44 

Reduction                          -  49 

Compound  Additon             -  56 

Compound  Substraction             -  61 

Compound  Multiplication            -  65 

Compound  Division             ...  73 

Exchange                         -  77 

Vulgar  Fractions            •  80 

Decimal  Fractions            -  97 

Duodecimals                           *  105 

Single  Rule  of  Three            -            -  108 

Double  Rule  of  Three            -  115 

Practice             -             -             -  119 

Fellowship             ->             ...  124 

Tare  and  Tret                          -             -  126 

Simple  Interest            ...  130 

Compound  Interest            -  137 

Insurance,  Commission  and  Brokerage  141 

Discount            -            -            -  142 

Equation                          .         .         „  143 

Loss  and  Gain            •        •  144 

Involution        --..:.  145 

Evolution        -  146 

Square  Root            -            •  146 

Cube  Root            -            -  149 

Alligation            -            •  151 

Single  Position            -                         .  154 
Double  Position            .            -            .156 

Arethmetical  Progression            *        -  157 

Geometrical  Progreesion                     -  160 

Permutation            -            -            -  162 

Mensuration            .            .            „  162 

Practical  Questions                         -  168 

Book-Keeping            -            -             -  171 


(*<* 


YA  02427 


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